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Chapter 4: Problem 40

Graph \(f(x)=5^{x}\) and \(g(x)=\log _{5} x\) in the same rectangular coordinate system.

Short Answer

Expert verified
Plot the graphs \(f(x)=5^{x}\) that sharply increases for positive x-values, and \(g(x)=\log _{5} x\) that slowly increases for positive x-values.

Step by step solution

01

Identify the points

Choose some convenient values of x, substitute them into the function, and calculate the corresponding function values. For \(f(x)=5^{x}\), let's take x = -1, 0, and 1. For \(g(x)=\log _{5} x\), it's important to choose x-values that are powers of 5 because we can easily compute the logarithms
02

Plot the points and draw the graph for \(f(x)=5^{x}\)

After calculating the corresponding function values, plot these points on the same rectangular coordinate grid and draw a smooth curve through the points. This graph is an exponential growth function, increasing rapidly for positive x-values
03

Plot the points and draw the graph for \(g(x)=\log _{5} x\)

As in step 2, after computing the corresponding function values, plot these points on the same rectangular coordinate grid and draw a smooth curve through the points. This graph is the logarithmic function which is the inverse of the exponential growth function. It slowly increases for positive x-values but does not exist for negative x-values (since you can't take a logarithm of a negative number)

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Most popular questions from this chapter

In Exercises \(1-40,\) use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \log (1000 x) $$

In Exercises \(1-40,\) use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \log N^{-6} $$

In Exercises \(1-40,\) use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \log _{9}(9 x) $$

In Exercises \(1-40,\) use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \log _{6}\left(\frac{36}{\sqrt{x+1}}\right) $$

In Exercises \(71-78,\) use common logarithms or natural logarithms and a calculator to evaluate to four decimal places. $$ \log _{5} 13 $$
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