Learning Outcomes
- Convert from logarithmic to exponential form.
- Convert from exponential to logarithmic form.
- Solve real-world problems of logarithmic functions.
In order to analyze the magnitude of earthquakes or compare the magnitudes of two different earthquakes, we need to be able to convert between logarithmic and exponential form. For example, suppose the amount of energy released from one earthquake was 500 times greater than the amount of energy released from another. We want to calculate the difference in magnitude. The equation that represents this problem is [latex]{10}^{x}=500[/latex] where xrepresents the difference in magnitudes on the Richter Scale. How would we solve forx?
We have not yet learned a method for solving exponential equations algebraically. None of the algebraic tools discussed so far is sufficient to solve [latex]{10}^{x}=500[/latex]. We know that [latex]{10}^{2}=100[/latex] and [latex]{10}^{3}=1000[/latex], so it is clear that xmust be some value between 2 and 3 since [latex]y={10}^{x}[/latex] is increasing. We can examine a graphto better estimate the solution.
Estimating from a graph, however, is imprecise. To find an algebraic solution, we must introduce a new function. Observe that the graph abovepasses the horizontal line test. The exponential function [latex]y={b}^{x}[/latex] is one-to-one, so its inverse, [latex]x={b}^{y}[/latex] is also a function. As is the case with all inverse functions, we simply interchange xand yand solve for yto find the inverse function. To represent yas a function of x, we use a logarithmic function of the form [latex]y={\mathrm{log}}_{b}\left(x\right)[/latex]. The base blogarithm of a number is the exponent by which we must raise bto get that number.
tip for success
Understanding what a logarithm is requires understanding what an exponent is. A logarithm is an exponent.
Read the paragraphs and boxes below carefully, perhaps more than once or twice, to gain the understanding of the inverse relationship between logarithms and exponents. Keep in mind that the inverse of a function effectively “undoes” what the other does.
You can use the definition of the logarithm given below to solve certain equations involving exponents and logarithms.
We read a logarithmic expression as, “The logarithm with base bof xis equal to y,” or, simplified, “log base bof xis y.” We can also say, “braised to the power of yis x,” because logs are exponents. For example, the base 2 logarithm of 32 is 5, because 5 is the exponent we must apply to 2 to get 32. Since [latex]{2}^{5}=32[/latex], we can write [latex]{\mathrm{log}}_{2}32=5[/latex]. We read this as “log base 2 of 32 is 5.”
We can express the relationship between logarithmic form and its corresponding exponential form as follows:
[latex]{\mathrm{log}}_{b}\left(x\right)=y\Leftrightarrow {b}^{y}=x,\text{}b>0,b\ne 1[/latex]
Note that the base bis always positive.
Because a logarithm is a function, it is most correctly written as [latex]{\mathrm{log}}_{b}\left(x\right)[/latex] using parentheses to denote function evaluation just as we would with [latex]f\left(x\right)[/latex]. However, when the input is a single variable or number, it is common to see the parentheses dropped and the expression written without parentheses as [latex]{\mathrm{log}}_{b}x[/latex]. Note that many calculators require parentheses around the x.
We can illustrate the notation of logarithms as follows:
Notice that when comparing the logarithm function and the exponential function, the input and the output are switched. This means [latex]y={\mathrm{log}}_{b}\left(x\right)[/latex] and [latex]y={b}^{x}[/latex] are inverse functions.
A General Note: Definition of the Logarithmic Function
A logarithm base bof a positive number xsatisfies the following definition:
For [latex]x>0,b>0,b\ne 1[/latex],
[latex]y={\mathrm{log}}_{b}\left(x\right)\text{ is equal to }{b}^{y}=x[/latex], where
- we read [latex]{\mathrm{log}}_{b}\left(x\right)[/latex] as, “the logarithm with base bof x” or the “log base bof x.”
- the logarithm yis the exponent to which bmust be raised to get x.
- if no base [latex]b[/latex] is indicated, the base of the logarithm is assumed to be [latex]10[/latex].
Also, since the logarithmic and exponential functions switch the xand yvalues, the domain and range of the exponential function are interchanged for the logarithmic function. Therefore,
- the domain of the logarithm function with base [latex]b \text{ is} \left(0,\infty \right)[/latex].
- the range of the logarithm function with base [latex]b \text{ is} \left(-\infty ,\infty \right)[/latex].
Q & A
Can we take the logarithm of a negative number?
No. Because the base of an exponential function is always positive, no power of that base can ever be negative. We can never take the logarithm of a negative number. Also, we cannot take the logarithm of zero. Calculators may output a log of a negative number when in complex mode, but the log of a negative number is not a real number.
How To: Given an equation in logarithmic form [latex]{\mathrm{log}}_{b}\left(x\right)=y[/latex], convert it to exponential form
- Examine the equation [latex]y={\mathrm{log}}_{b}x[/latex] and identify b, y, and x.
- Rewrite [latex]{\mathrm{log}}_{b}x=y[/latex] as [latex]{b}^{y}=x[/latex].
Example: Converting from Logarithmic Form to Exponential Form
Write the following logarithmic equations in exponential form.
- [latex]{\mathrm{log}}_{6}\left(\sqrt{6}\right)=\frac{1}{2}[/latex]
- [latex]{\mathrm{log}}_{3}\left(9\right)=2[/latex]
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Write the following logarithmic equations in exponential form.
- [latex]{\mathrm{log}}_{10}\left(1,000,000\right)=6[/latex]
- [latex]{\mathrm{log}}_{5}\left(25\right)=2[/latex]
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Convert from Exponential to Logarithmic Form
To convert from exponential to logarithmic form, we follow the same steps in reverse. We identify the base b, exponent x, and output y. Then we write [latex]x={\mathrm{log}}_{b}\left(y\right)[/latex].
Example: Converting from Exponential Form to Logarithmic Form
Write the following exponential equations in logarithmic form.
- [latex]{2}^{3}=8[/latex]
- [latex]{5}^{2}=25[/latex]
- [latex]{10}^{-4}=\frac{1}{10,000}[/latex]
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Write the following exponential equations in logarithmic form.
- [latex]{3}^{2}=9[/latex]
- [latex]{5}^{3}=125[/latex]
- [latex]{2}^{-1}=\frac{1}{2}[/latex]
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Real-World Applications
Example: Rewriting and Solving a Real-World Exponential Model
The amount of energy released from one earthquake was [latex]500[/latex] times greater than the amount of energy released from another. The equation [latex]10^x=500[/latex]represents this situation, where [latex]x[/latex]is the difference in magnitudes on the Richter Scale. To the nearest thousandth, what was the difference in magnitudes?
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The amount of energy released from one earthquake was [latex]8,500[/latex]times greater than the amount of energy released from another. The equation [latex]10^x=8500[/latex]represents this situation, where [latex]x[/latex]is the difference in magnitudes on the Richter Scale. To the nearest thousandth, what was the difference in magnitudes?
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Example: Exposure Index EI
The exposure index EIfor a 35-millimeter camera is a measurement of the amount of light that hits the film. It is determined by the equation [latex]EI = log_2(\frac{f^2}{t})[/latex], where [latex]f[/latex] is the “f-stop” setting on the camera, and [latex]t[/latex] is the exposure time in seconds. Suppose the f-stop setting is 8 and the desired exposure time is 2 seconds. What will the resulting exposure index be?
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Refer to the previous example. Suppose the light meter on a camera indicates anEIof [latex]-2[/latex],and the desired exposure time is 16 seconds. What should the f-stop setting be? (* Hint: f-stop is a positive number.)
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If 1 in [latex]x[/latex] person dies as a result of doing some given activity each year, the safety index for that activity is [latex]SI = log x[/latex]. For example, according to a statistic in the US, 1 in 5,300 dies each year due to car crashes. Then the safety index for car crash is [latex]log(5300) \approx 3.7[/latex]. If 1 in 2,000,000 is killed by lighting, what is the safety index for lightning?[1]
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