4.7: Exponential and Logarithmic Equations (2024)

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    Learning Objectives
    • Use like bases to solve exponential equations.
    • Use logarithms to solve exponential equations.
    • Use the definition of a logarithm to solve logarithmic equations.
    • Use the one-to-one property of logarithms to solve logarithmic equations.
    • Solve applied problems involving exponential and logarithmic equations.

    In 1859, an Australian landowner named Thomas Austin released \(24\) rabbits into the wild for hunting. Because Australia had few predators and ample food, the rabbit population exploded. In fewer than ten years, the rabbit population numbered in the millions.

    4.7: Exponential and Logarithmic Equations (2)

    Uncontrolled population growth, as in the wild rabbits in Australia, can be modeled with exponential functions. Equations resulting from those exponential functions can be solved to analyze and make predictions about exponential growth. In this section, we will learn techniques for solving exponential functions.

    Using Like Bases to Solve Exponential Equations

    The first technique involves two functions with like bases. Recall that the one-to-one property of exponential functions tells us that, for any real numbers \(b\), \(S\), and \(T\), where \(b>0\), \(b≠1\), \(b^S=b^T\) if and only if \(S=T\).

    In other words, when an exponential equation has the same base on each side, the exponents must be equal. This also applies when the exponents are algebraic expressions. Therefore, we can solve many exponential equations by using the rules of exponents to rewrite each side as a power with the same base. Then, we use the fact that exponential functions are one-to-one to set the exponents equal to one another, and solve for the unknown.

    For example, consider the equation \(3^{4x−7}=\dfrac{3^{2x}}{3}\). To solve for \(x\), we use the division property of exponents to rewrite the right side so that both sides have the common base, \(3\). Then we apply the one-to-one property of exponents by setting the exponents equal to one another and solving for \(x\):

    \[\begin{align*} 3^{4x-7}&= \dfrac{3^{2x}}{3}\\ 3^{4x-7}&= \dfrac{3^{2x}}{3^1} \qquad &&\text{Rewrite 3 as } 3^1\\ 3^{4x-7}&= 3^{2x-1} \qquad &&\text{Use the division property of exponents}\\ 4x-7&= 2x-1 \qquad &&\text{Apply the one-to-one property of exponents}\\ 2x&= 6 \qquad &&\text{Subtract 2x and add 7 to both sides}\\ x&= 3 \qquad &&\text{Divide by 3} \end{align*}\]

    USING THE ONE-TO-ONE PROPERTY OF EXPONENTIAL FUNCTIONS TO SOLVE EXPONENTIAL EQUATIONS

    For any algebraic expressions \(S\) and \(T\), and any positive real number \(b≠1\),

    \[\begin{align} b^S=b^T\text{ if and only if } S=T \end{align}\]

    How to: Given an exponential equation with the form \(b^S=b^T\), where \(S\) and \(T\) are algebraic expressions with an unknown, solve for the unknown.
    1. Use the rules of exponents to simplify, if necessary, so that the resulting equation has the form \(b^S=b^T\).
    2. Use the one-to-one property to set the exponents equal.
    3. Solve the resulting equation, \(S=T\), for the unknown.
    Example \(\PageIndex{1}\): Solving an Exponential Equation with a Common Base

    Solve \(2^{x−1}=2^{2x−4}\).

    Solution

    \[\begin{align*} 2^{x-1}&= 2^{2x-4} \qquad &&\text{The common base is 2}\\ x-1&= 2x-4 \qquad &&\text{By the one-to-one property the exponents must be equal}\\ x&= 3 \qquad &&\text{Solve for x} \end{align*}\]

    Exercise \(\PageIndex{1}\)

    Solve \(5^{2x}=5^{3x+2}\).

    Answer

    \(x=−2\)

    Rewrite Equations So All Powers Have the Same Base

    Sometimes the common base for an exponential equation is not explicitly shown. In these cases, we simply rewrite the terms in the equation as powers with a common base, and solve using the one-to-one property.

    For example, consider the equation \(256=4^{x−5}\). We can rewrite both sides of this equation as a power of \(2\). Then we apply the rules of exponents, along with the one-to-one property, to solve for \(x\):

    \[\begin{align*} 256&= 4^{x-5}\\ 2^8&= {(2^2)}^{x-5} \qquad &&\text{Rewrite each side as a power with base 2}\\ 2^8&= 2^{2x-10} \qquad &&\text{Use the one-to-one property of exponents}\\ 8&= 2x-10 \qquad &&\text{Apply the one-to-one property of exponents}\\ 18&= 2x \qquad &&\text{Add 10 to both sides}\\ x&= 9 \qquad &&\text{Divide by 2} \end{align*}\]

    How to: Given an exponential equation with unlike bases, use the one-to-one property to solve it.
    1. Rewrite each side in the equation as a power with a common base.
    2. Use the rules of exponents to simplify, if necessary, so that the resulting equation has the form \(b^S=b^T\).
    3. Use the one-to-one property to set the exponents equal.
    4. Solve the resulting equation, \(S=T\), for the unknown.
    Example \(\PageIndex{2}\): Solving Equations by Rewriting Them to Have a Common Base

    Solve \(8^{x+2}={16}^{x+1}\).

    Solution

    \[\begin{align*} 8^{x+2}&= {16}^{x+1}\\ {(2^3)}^{x+2}&= {(2^4)}^{x+1} \qquad &&\text{Write 8 and 16 as powers of 2}\\ 2^{3x+6}&= 2^{4x+4} \qquad &&\text{To take a power of a power, multiply exponents}\\ 3x+6&= 4x+4 \qquad &&\text{Use the one-to-one property to set the exponents equal}\\ x&= 2 \qquad &&\text{Solve for x} \end{align*}\]

    Exercise \(\PageIndex{2}\)

    Solve \(5^{2x}={25}^{3x+2}\).

    Answer

    \(x=−1\)

    Example \(\PageIndex{3}\): Solving Equations by Rewriting Roots with Fractional Exponents to Have a Common Base

    Solve \(2^{5x}=\sqrt{2}\).

    Solution

    \[\begin{align*} 2^{5x}&= 2^{\frac{1}{2}} \qquad &&\text{Write the square root of 2 as a power of 2}\\ 5x&= \dfrac{1}{2} \qquad &&\text{Use the one-to-one property}\\ x&= \dfrac{1}{10} \qquad &&\text{Solve for x} \end{align*}\]

    Exercise \(\PageIndex{3}\)

    Solve \(5^x=\sqrt{5}\).

    Answer

    \(x=\dfrac{1}{2}\)

    Q&A: Do all exponential equations have a solution? If not, how can we tell if there is a solution during the problem-solving process?

    No. Recall that the range of an exponential function is always positive. While solving the equation, we may obtain an expression that is undefined.

    Example \(\PageIndex{4}\): Solving an Equation with Positive and Negative Powers

    Solve \(3^{x+1}=−2\).

    Solution

    This equation has no solution. There is no real value of \(x\) that will make the equation a true statement because any power of a positive number is positive.

    Analysis

    Figure \(\PageIndex{2}\) shows that the two graphs do not cross so the left side is never equal to the right side. Thus the equation has no solution.

    4.7: Exponential and Logarithmic Equations (3)
    Exercise \(\PageIndex{4}\)

    Solve \(2^x=−100\).

    Answer

    The equation has no solution.

    Solving Exponential Equations Using Logarithms

    Sometimes the terms of an exponential equation cannot be rewritten with a common base. In these cases, we solve by taking the logarithm of each side. Recall, since \(\log(a)=\log(b)\) is equivalent to \(a=b\), we may apply logarithms with the same base on both sides of an exponential equation.

    How to: Given an exponential equation in which a common base cannot be found, solve for the unknown
    1. Apply the logarithm of both sides of the equation.
      • If one of the terms in the equation has base 10, use the common logarithm.
      • If none of the terms in the equation has base 10, use the natural logarithm.
    2. Use the rules of logarithms to solve for the unknown.
    Example \(\PageIndex{5}\): Solving an Equation Containing Powers of Different Bases

    Solve \(5^{x+2}=4^x\).

    Solution

    \[\begin{align*}
    5^{x+2}&= 4^x \qquad &&\text{There is no easy way to get the powers to have the same base}\\
    \ln5^{x+2}&= \ln4^x \qquad &&\text{Take ln of both sides}\\
    (x+2)\ln5&= x\ln4 \qquad &&\text{Use laws of logs}\\
    x\ln5+2\ln5&= x\ln4 \qquad &&\text{Use the distributive law}\\
    x\ln5-x\ln4&= -2\ln5 \qquad &&\text{Get terms containing x on one side, terms without x on the other}\\
    x(\ln5-\ln4)&= -2\ln5 \qquad &&\text{On the left hand side, factor out an x}\\
    x\ln \left (\dfrac{5}{4} \right )&= \ln \left (\dfrac{1}{25} \right ) \qquad &&\text{Use the laws of logs}\\
    x&=\dfrac{\ln \left (\dfrac{1}{25} \right )}{\ln \left (\dfrac{5}{4} \right )} \qquad &&\text{Divide by the coefficient of x}
    \end{align*}\]

    Exercise \(\PageIndex{5}\)

    Solve \(2^x=3^{x+1}\).

    Answer

    \(x=\dfrac{\ln3}{\ln \left (\dfrac{2}{3} \right )}\)

    Q&A: Is there any way to solve \(2^x=3^x\)?

    Yes. The solution is \(0\).

    Equations Containing \(e\)

    One common type of exponential equations are those with base \(e\). This constant occurs again and again in nature, in mathematics, in science, in engineering, and in finance. When we have an equation with a base \(e\) on either side, we can use the natural logarithm to solve it.

    How to: Given an equation of the form \(y=Ae^{kt}\), solve for \(t\).
    1. Divide both sides of the equation by \(A\).
    2. Apply the natural logarithm of both sides of the equation.
    3. Divide both sides of the equation by \(k\).
    Example \(\PageIndex{6}\): Solve an Equation of the Form \(y = Ae^{kt}\)

    Solve \(100=20e^{2t}\).

    Solution

    \[\begin{align*} 100&= 20e^{2t}\\ 5&= e^{2t} \qquad &&\text{Divide by the coefficient of the power}\\ \ln5&= 2t \qquad &&\text{Take ln of both sides. Use the fact that } ln(x) \text{ and } e^x \text{ are inverse functions}\\ t&= \dfrac{\ln5}{2} \qquad &&\text{Divide by the coefficient of t} \end{align*}\]

    Analysis

    Using laws of logs, we can also write this answer in the form \(t=\ln\sqrt{5}\). If we want a decimal approximation of the answer, we use a calculator.

    Exercise \(\PageIndex{6}\)

    Solve \(3e^{0.5t}=11\).

    Answer

    \(t=2\ln \left (\dfrac{11}{3} \right )\) or \(\ln{ \left (\dfrac{11}{3} \right )}^2\)

    Q&A: Does every equation of the form \(y=Ae^{kt}\) have a solution?

    No. There is a solution when \(k≠0\),and when \(y\) and \(A\) are either both 0 or neither 0, and they have the same sign. An example of an equation with this form that has no solution is \(2=−3e^t\).

    Example \(\PageIndex{7}\): Solving an Equation That Can Be Simplified to the Form \(y=Ae^{kt}\)

    Solve \(4e^{2x}+5=12\).

    Solution

    \[\begin{align*} 4e^{2x}+5&= 12\\ 4e^{2x}&= 7 \qquad &&\text{Combine like terms}\\ e^{2x}&= \dfrac{7}{4} \qquad &&\text{Divide by the coefficient of the power}\\ 2x&= \ln \left (\dfrac{7}{4} \right ) \qquad &&\text{Take ln of both sides}\\ x&= \dfrac{1}{2}\ln \left (\dfrac{7}{4} \right ) \qquad &&\text{Solve for x} \end{align*}\]

    Exercise \(\PageIndex{7}\)

    Solve \(3+e^{2t}=7e^{2t}\).

    Answer

    \(t=\ln \left (\dfrac{1}{\sqrt{2}} \right )=−\dfrac{1}{2}\ln(2)\)

    Extraneous Solutions

    Sometimes the methods used to solve an equation introduce an extraneous solution, which is a solution that is correct algebraically but does not satisfy the conditions of the original equation. One such situation arises in solving when the logarithm is taken on both sides of the equation. In such cases, remember that the argument of the logarithm must be positive. If the number we are evaluating in a logarithm function is negative, there is no output.

    Example \(\PageIndex{8}\): Solving Exponential Functions in Quadratic Form

    Solve \(e^{2x}−e^x=56\).

    Solution

    \[\begin{align*} e^{2x}-e^x&= 56\\ e^{2x}-e^x-56&= 0 \qquad &&\text{Get one side of the equation equal to zero}\\ (e^x+7)(e^x-8)&= 0 \qquad &&\text{Factor by the FOIL method}\\ e^x+7&= 0 \qquad &&\text{or} \\ e^x-8&= 0 \qquad &&\text{If a product is zero, then one factor must be zero}\\ e^x&= -7 \qquad &&\text{or} \\ e^x&= 8 \qquad &&\text{Isolate the exponentials}\\ e^x&= 8 \qquad &&\text{Reject the equation in which the power equals a negative number}\\ x&= \ln8 \qquad &&\text{Solve the equation in which the power equals a positive number} \end{align*}\]

    Analysis

    When we plan to use factoring to solve a problem, we always get zero on one side of the equation, because zero has the unique property that when a product is zero, one or both of the factors must be zero. We reject the equation \(e^x=−7\) because a positive number never equals a negative number. The solution \(\ln(−7)\) is not a real number, and in the real number system this solution is rejected as an extraneous solution.

    Exercise \(\PageIndex{8}\)

    Solve \(e^{2x}=e^x+2\).

    Answer

    \(x=\ln2\)

    Q&A: Does every logarithmic equation have a solution?

    No. Keep in mind that we can only apply the logarithm to a positive number. Always check for extraneous solutions.

    Using the Definition of a Logarithm to Solve Logarithmic Equations

    We have already seen that every logarithmic equation \({\log}_b(x)=y\) is equivalent to the exponential equation \(b^y=x\). We can use this fact, along with the rules of logarithms, to solve logarithmic equations where the argument is an algebraic expression.

    For example, consider the equation \({\log}_2(2)+{\log}_2(3x−5)=3\). To solve this equation, we can use rules of logarithms to rewrite the left side in compact form and then apply the definition of logs to solve for \(x\):

    \[\begin{align*} {\log}_2(2)+{\log}_2(3x-5)&= 3\\ {\log}_2(2(3x-5))&= 3 \qquad \text{Apply the product rule of logarithms}\\ {\log}_2(6x-10)&= 3 \qquad \text{Distribute}\\ 2^3&= 6x-10 \qquad \text{Apply the definition of a logarithm}\\ 8&= 6x-10 \qquad \text{Calculate } 2^3\\ 18&= 6x \qquad \text{Add 10 to both sides}\\ x&= 3 \qquad \text{Divide by 6} \end{align*}\]

    USING THE DEFINITION OF A LOGARITHM TO SOLVE LOGARITHMIC EQUATIONS

    For any algebraic expression \(S\) and real numbers \(b\) and \(c\),where \(b>0\), \(b≠1\),

    \[\begin{align} {\log}_b(S)=c \text{ if and only if } b^c=S \end{align}\]

    Example \(\PageIndex{9}\): Using Algebra to Solve a Logarithmic Equation

    Solve \(2\ln x+3=7\).

    Solution

    \[\begin{align*} 2\ln x+3&= 7\\ 2\ln x&= 4 \qquad \text{Subtract 3}\\ \ln x&= 2 \qquad \text{Divide by 2}\\ x&= e^2 \qquad \text{Rewrite in exponential form} \end{align*}\]

    Exercise \(\PageIndex{9}\)

    Solve \(6+\ln x=10\).

    Answer

    \(x=e^4\)

    Example \(\PageIndex{10}\): Using Algebra Before and After Using the Definition of the Natural Logarithm

    Solve \(2\ln(6x)=7\).

    Solution

    \[\begin{align*} 2\ln(6x)&= 7\\ \ln(6x)&= \dfrac{7}{2} \qquad \text{Divide by 2}\\ 6x&= e^{\left (\dfrac{7}{2} \right )} \qquad \text{Use the definition of }\ln \\ x&= \dfrac{1}{6}e^{\left (\dfrac{7}{2} \right )} \qquad \text{Divide by 6} \end{align*}\]

    Exercise \(\PageIndex{10}\)

    Solve \(2\ln(x+1)=10\).

    Answer

    \(x=e^5−1\)

    Example \(\PageIndex{11}\): Using a Graph to Understand the Solution to a Logarithmic Equation

    Solve \(\ln x=3\).

    Solution

    \[\begin{align*} \ln x&= 3\\ x&= e^3 \qquad \text{Use the definition of the natural logarithm} \end{align*}\]

    Figure \(\PageIndex{3}\) represents the graph of the equation. On the graph, the x-coordinate of the point at which the two graphs intersect is close to \(20\). In other words \(e^3≈20\). A calculator gives a better approximation: \(e^3≈20.0855\).

    4.7: Exponential and Logarithmic Equations (4)
    Exercise \(\PageIndex{11}\)

    Use a graphing calculator to estimate the approximate solution to the logarithmic equation \(2^x=1000\) to \(2\) decimal places.

    Answer

    \(x≈9.97\)

    Using the One-to-One Property of Logarithms to Solve Logarithmic Equations

    As with exponential equations, we can use the one-to-one property to solve logarithmic equations. The one-to-one property of logarithmic functions tells us that, for any real numbers \(x>0\), \(S>0\), \(T>0\) and any positive real number \(b\), where \(b≠1\),

    \({\log}_bS={\log}_bT\) if and only if \(S=T\).

    For example,

    If \({\log}_2(x−1)={\log}_2(8)\), then \(x−1=8\).

    So, if \(x−1=8\), then we can solve for \(x\),and we get \(x=9\). To check, we can substitute \(x=9\) into the original equation: \({\log}_2(9−1)={\log}_2(8)=3\). In other words, when a logarithmic equation has the same base on each side, the arguments must be equal. This also applies when the arguments are algebraic expressions. Therefore, when given an equation with logs of the same base on each side, we can use rules of logarithms to rewrite each side as a single logarithm. Then we use the fact that logarithmic functions are one-to-one to set the arguments equal to one another and solve for the unknown.

    For example, consider the equation \(\log(3x−2)−\log(2)=\log(x+4)\). To solve this equation, we can use the rules of logarithms to rewrite the left side as a single logarithm, and then apply the one-to-one property to solve for \(x\):

    \[\begin{align*} \log(3x-2)-\log(2)&= \log(x+4)\\ \log \left (\dfrac{3x-2}{2} \right )&= \log(x+4) \qquad \text{Apply the quotient rule of logarithms}\\ \dfrac{3x-2}{2}&= x+4 \qquad \text{Apply the one to one property of a logarithm}\\ 3x-2&= 2x+8 \qquad \text{Multiply both sides of the equation by 2}\\ x&= 10 \qquad \text{Subtract 2x and add 2} \end{align*}\]

    To check the result, substitute \(x=10\) into \(\log(3x−2)−\log(2)=\log(x+4)\).

    \[\begin{align*} \log(3(10)-2)-\log(2)&= \log((10)+4) \\ \log(28)-\log(2)&= \log(14)\\ \log \left (\dfrac{28}{2} \right )&= \log(14) \qquad \text{The solution checks} \end{align*}\]

    USING THE ONE-TO-ONE PROPERTY OF LOGARITHMS TO SOLVE LOGARITHMIC EQUATIONS

    For any algebraic expressions \(S\) and \(T\) and any positive real number \(b\), where \(b≠1\),

    \[\begin{align} b^S=b^T\text{ if and only if } S=T \end{align}\]

    Note, when solving an equation involving logarithms, always check to see if the answer is correct or if it is an extraneous solution.

    How to: Given an equation containing logarithms, solve it using the one-to-one property
    1. Use the rules of logarithms to combine like terms, if necessary, so that the resulting equation has the form \({\log}_bS={\log}_bT\).
    2. Use the one-to-one property to set the arguments equal.
    3. Solve the resulting equation, \(S=T\), for the unknown.
    Example \(\PageIndex{12}\): Solving an Equation Using the One-to-One Property of Logarithms

    Solve \(\ln(x^2)=\ln(2x+3)\).

    Solution

    \[\begin{align*} \ln(x^2)&= \ln(2x+3)\\ x^2&= 2x+3 \qquad \text{Use the one-to-one property of the logarithm}\\ x^2-2x-3&= 0 \qquad \text{Get zero on one side before factoring}\\ (x-3)(x+1)&= 0 \qquad \text{Factor using FOIL}\\ x-3&= 0 \qquad \text{or } x+1=0 \text{ If a product is zero, one of the factors must be zero}\\ x=3 \qquad \text{or} \\ x&= -11 \qquad \text{Solve for x} \end{align*}\]

    Analysis

    There are two solutions: \(3\) or \(−1\). The solution \(−1\) is negative, but it checks when substituted into the original equation because the argument of the logarithm functions is still positive.

    Exercise \(\PageIndex{12}\)

    Solve \(\ln(x^2)=\ln1\).

    Answer

    \(x=1\) or \(x=−1\)

    Solving Applied Problems Using Exponential and Logarithmic Equations

    In previous sections, we learned the properties and rules for both exponential and logarithmic functions. We have seen that any exponential function can be written as a logarithmic function and vice versa. We have used exponents to solve logarithmic equations and logarithms to solve exponential equations. We are now ready to combine our skills to solve equations that model real-world situations, whether the unknown is in an exponent or in the argument of a logarithm.

    One such application is in science, in calculating the time it takes for half of the unstable material in a sample of a radioactive substance to decay, called its half-life. Table \(\PageIndex{1}\) lists the half-life for several of the more common radioactive substances.

    Table \(\PageIndex{1}\)
    Substance Use Half-life
    gallium-67 nuclear medicine 80 hours
    cobalt-60 manufacturing 5.3 years
    technetium-99m nuclear medicine 6 hours
    americium-241 construction 432 years
    carbon-14 archeological dating 5,715 years
    uranium-235 atomic power 703,800,000 years

    We can see how widely the half-lives for these substances vary. Knowing the half-life of a substance allows us to calculate the amount remaining after a specified time. We can use the formula for radioactive decay:

    \[\begin{align} A(t)&= A_0e^{\tfrac{\ln(0.5)}{T}t}\\ A(t)&= A_0e^{\tfrac{\ln(0.5)t}{T}}\\ A(t)&= A_0{(e^{\ln(0.5)})}^{\tfrac{t}{T}}\\ A(t)&= A_0{\left (\dfrac{1}{2}\right )}^{\tfrac{t}{T}}\\ \end{align}\]

    where

    • \(A_0\) is the amount initially present
    • \(T\) is the half-life of the substance
    • \(t\) is the time period over which the substance is studied
    • \(y\) is the amount of the substance present after time \(t\)
    Example \(\PageIndex{13}\): Using the Formula for Radioactive Decay to Find the Quantity of a Substance

    How long will it take for ten percent of a \(1000\)-gram sample of uranium-235 to decay?

    Solution

    \[\begin{align*} y&= 1000e^{\tfrac{\ln(0.5)}{703,800,000}t}\\ 900&= 1000e^{\tfrac{\ln(0.5)}{703,800,000}t} \qquad \text{After } 10\% \text{ decays, 900 grams are left}\\ 0.9&= e^{\tfrac{\ln(0.5)}{703,800,000}t} \qquad \text{Divide by 1000}\\ \ln(0.9)&= \ln \left (e^{\tfrac{\ln(0.5)}{703,800,000}t} \right ) \qquad \text{Take ln of both sides}\\ \ln(0.9)&= \dfrac{\ln(0.5)}{703,800,000}t \qquad \ln(e^M)=M\\ t&= 703,800,000\times \dfrac{\ln(0.9)}{\ln(0.5)} \qquad \text{years Solve for t}\\ t&\approx 106,979,777 \qquad \text{years} \end {align*} \]

    Analysis

    Ten percent of \(1000\) grams is \(100\) grams. If \(100\) grams decay, the amount of uranium-235 remaining is \(900\) grams.

    Exercise \(\PageIndex{13}\)

    How long will it take before twenty percent of our 1000-gram sample of uranium-235 has decayed?

    Answer

    \(t=703,800,000×\dfrac{\ln(0.8)}{\ln(0.5)}\)years ≈ 226,572,993 years.

    Media

    Access these online resources for additional instruction and practice with exponential and logarithmic equations.

    • Solving Logarithmic Equations
    • Solving Exponential Equations with Logarithms

    Key Equations

    One-to-one property for exponential functions For any algebraic expressions \(S\) and \(T\) and any positive real number \(b\), where \(b^S=b^T\) if and only if \(S=T\).
    Definition of a logarithm For any algebraic expression S and positive real numbers \(b\) and \(c\), where \(b≠1\), \({\log}_b(S)=c\) if and only if \(b^c=S\).
    One-to-one property for logarithmic functions For any algebraic expressions \(S\) and \(T\) and any positive real number \(b\), where \(b≠1\),
    \({\log}_bS={\log}_bT\) if and only if \(S=T\).

    Key Concepts

    • We can solve many exponential equations by using the rules of exponents to rewrite each side as a power with the same base. Then we use the fact that exponential functions are one-to-one to set the exponents equal to one another and solve for the unknown.
    • When we are given an exponential equation where the bases are explicitly shown as being equal, set the exponents equal to one another and solve for the unknown. See Example \(\PageIndex{1}\).
    • When we are given an exponential equation where the bases are not explicitly shown as being equal, rewrite each side of the equation as powers of the same base, then set the exponents equal to one another and solve for the unknown. See Example \(\PageIndex{2}\), Example \(\PageIndex{3}\), and Example \(\PageIndex{4}\).
    • When an exponential equation cannot be rewritten with a common base, solve by taking the logarithm of each side. See Example \(\PageIndex{5}\).
    • We can solve exponential equations with base \(e\),by applying the natural logarithm of both sides because exponential and logarithmic functions are inverses of each other. See Example \(\PageIndex{6}\) and Example \(\PageIndex{7}\).
    • After solving an exponential equation, check each solution in the original equation to find and eliminate any extraneous solutions. See Example \(\PageIndex{8}\).
    • When given an equation of the form \({\log}_b(S)=c\), where \(S\) is an algebraic expression, we can use the definition of a logarithm to rewrite the equation as the equivalent exponential equation \(b^c=S\), and solve for the unknown. See Example \(\PageIndex{9}\) and Example \(\PageIndex{10}\).
    • We can also use graphing to solve equations with the form \({\log}_b(S)=c\). We graph both equations \(y={\log}_b(S)\) and \(y=c\) on the same coordinate plane and identify the solution as the x-value of the intersecting point. See Example \(\PageIndex{11}\).
    • When given an equation of the form \({\log}_bS={\log}_bT\), where \(S\) and \(T\) are algebraic expressions, we can use the one-to-one property of logarithms to solve the equation \(S=T\) for the unknown. See Example \(\PageIndex{12}\).
    • Combining the skills learned in this and previous sections, we can solve equations that model real world situations, whether the unknown is in an exponent or in the argument of a logarithm. See Example \(\PageIndex{13}\).
    4.7: Exponential and Logarithmic Equations (2024)

    FAQs

    What are exponential equations and logarithmic equations? ›

    An exponential equation is an equation in which the variable appears in an exponent. A logarithmic equation is an equation that involves the logarithm of an expression containing a variable. To solve exponential equations, first see whether you can write both sides of the equation as powers of the same number.

    What is the formula for logs and exponents? ›

    The formula of log to exponential form is logaN=x l o g a N = x , is written in exponential form as ax=N a x = N . The logarithm of a number N to the base of a is equal to x, which if written in exponential form is equal to a to the exponent of x is equal to N.

    How to solve an exponential equation? ›

    Step 1: Isolate the exponential and then apply the logarithm to both sides. Step 2: Apply the power rule for logarithms and write the exponent as a factor of the base. Step 3: Solve the resulting equation.

    What is an example of exponential form? ›

    A number in exponential form is written by taking the base number used as a factor multiple times, and raising it to the power of the number of times it is used as a factor, for example, 3 to the fourth power is 3x3x3x3, or 3 multiplied by itself four times.

    How do you rewrite between logarithmic and exponential functions? ›

    Step 1: Use the general form of a logarithmic function l o g b ( M ) = N to identify the value for M, b, and N in your logarithmic function. Step 2: Use these values to write your equivalent exponential function. Substitute these values into the exponential general form: M = b N .

    How do you understand exponential and logarithmic functions? ›

    Logarithmic functions are the inverses of exponential functions. The inverse of the exponential function y = ax is x = ay. The logarithmic function y = logax is defined to be equivalent to the exponential equation x = ay. y = logax only under the following conditions: x = ay, a > 0, and a≠1.

    What are the rules for exponential and logarithmic functions? ›

    We saw earlier that an exponential function is any function of the form f(x)=bx, where b>0 and b≠1. A logarithmic function is any function of the form g(x)=logb(x), where b>0 and b≠1. It is no coincidence that both forms have the same restrictions on b because they are inverses of each other.

    How to convert between exponential and logarithmic form? ›

    The exponential form ax=N a x = N is converted to logarithmic form logaN=x l o g a N = x . The exponent form of a to the exponent of x is equal to N, which on converting to logarithmic form we have log of N to the base of a is equal to x.

    How to solve exponential equations using logarithms step? ›

    Step 1: Isolate the exponential expression. Step 2: Take the logarithm of both sides. In this case, we will take the common logarithm of both sides so that we can approximate our result on a calculator. Step 3: Apply the power rule for logarithms and then solve.

    How to write as an exponential equation? ›

    For linear equations, we have y = m (slope) x + b (y intercept) and for exponential equations we have y = a (initial value)*r(ratio or base)^x.

    What is the difference between logarithmic and exponential? ›

    An exponential function has the form ax, where a is a constant; examples are 2x, 10x, ex. The logarithmic functions are the inverses of the exponential functions, that is, functions that "undo'' the exponential functions, just as, for example, the cube root function "undoes'' the cube function: 3√23=2.

    What is a logarithmic equation with an example? ›

    Logarithms only take in positive numbers as a value. Given the equation ⁡ , this is like saying "b raised to the power of c equals a." Examples of logarithm equations include ⁡ x + 5 = 7 , ⁡ 5 x − log 5 ⁡ x − 3 = 1 , and 7 log 3 ⁡ ( 3 x + 1 ) + log 3 ⁡ ( 3 x ) = 14 .

    What is the definition of exponential equation with example? ›

    What is an Exponential Equation? An exponential equation can be easily recognized as an equation with a variable in the exponent position. An example of this is y = 2 x . The number that has the variable exponent is called the base.

    What is exponential and logarithmic equations and inequalities? ›

    Exponential inequalities are inequalities that can be written in the form abx < k, where < can also be >, ≤, or ≥. Logarithmic inequalities are inequalities that can be written in the form logbx < k, where < can also be >, ≤, or ≥.

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