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Chapter 3: Problem 15

Write the exponential equation in logarithmic form. For example, the logarithmic form of \(2^{3}=8\) is \(\log _{2} 8=3\). $$5^{3}=125$$

Short Answer

Expert verified
The logarithmic form of the equation \(5^{3}=125\) is \(\log_{5} 125=3.\)

Step by step solution

01

Identify the base, exponent and result

Identify the base, exponent and the result in the given exponential equation. For given equation \(5^{3}=125\), base is 5, exponent is 3 and result is 125.
02

Write the base as the base of logarithm

Using the base 5, begin to form the logarithm equation, which is \(\log_{5}\) at this point.
03

Add the result of the equation

Next, add the result of the exponential equation to the logarithmic equation, which becomes \(\log_{5} 125\).
04

Equating it to the exponent

Finally, set the equation equal to the exponent in the given equation. The log equation becomes \(\log_{5} 125=3\)

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Most popular questions from this chapter

The populations \(P\) (in thousands) of Reno, Nevada from 2000 through 2007 can be modeled by \(P=346.8 e^{k t},\) where \(t\) represents the year, with \(t=0\) corresponding to \(2000 .\) In \(2005,\) the population of Reno was about 395,000 . (Source: U.S. Census Bureau) (a) Find the value of \(k\). Is the population increasing or decreasing? Explain. (b) Use the model to find the populations of Reno in 2010 and 2015 . Are the results reasonable? Explain. (c) According to the model, during what year will the population reach \(500,000 ?\)

The management at a plastics factory has found that the maximum number of units a worker can produce in a day is \(30 .\) The learning curve for the number \(N\) of units produced per day after a new employee has worked \(t\) days is modeled by \(N=30\left(1-e^{k t}\right) .\) After 20 days on the job, a new employee produces 19 units. (a) Find the learning curve for this employee (first, find the value of \(k\) ). (b) How many days should pass before this employee is producing 25 units per day?

The IQ scores for a sample of a class of returning adult students at a small northeastern college roughly follow the normal distribution \(y=0.0266 e^{-(x-100)^{2} / 450}, 70 \leq x \leq 115,\) where \(x\) is the IQ score. (a) Use a graphing utility to graph the function. (b) From the graph in part (a), estimate the average IQ score of an adult student.

Use a graphing utility to graph and solve the equation. Approximate the result to three decimal places. Verify your result algebraically. $$2 \ln (x+3)=3$$

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