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Chapter 3: Problem 36

Sketch the graphs of \(f\) and \(g\) in the same coordinate plane. $$f(x)=10^{x}, g(x)=\log x$$

Short Answer

Expert verified
The graph of \( f(x) = 10^{x} \) is a curve that increases rapidly as \( x \) increases and the curve of \( g(x) = \log{x} \) increases slowly and approaches negative infinity as \( x \) moves towards 0 from the right. The two functions are inverses. Therefore, the graph of \( g(x) \) is a reflection of the graph of \( f(x) \) over the line \( y=x \).

Step by step solution

01

Understand the Functions

Two functions are given in the problem, exponential and logarithmic. It will be important to ensure that the basic nature of these functions on the graph is understood. For \(f(x)=10^{x}\), as \(x\) increases, \(f(x)\) grows very rapidly. It's always positive and approaches 0 as \(x\) approaches negative infinity. For \(g(x)=\log{x}\), it is undefined at \(x=0\), grows slowly as \(x\) increases, and approaches negative infinity as \(x\) approaches 0 from the right.
02

Plot \(f(x)=10^{x}\)

On a coordinate plane, begin by plotting some key points. At \(x=0\), \(f(0)=10^{0}=1\). And when\(x=1, f(1)=10\), and at \(x=-1, f(-1)=0.1\). Plot these points and then sketch the typical shape of an exponential function.
03

Plot \(g(x)=\log{x}\)

When plotting the function \(g(x)=log{x}\), keep in mind that it is the inverse of the function \(f(x)=10^{x}\). Thus, the plot of \(g(x)\) will be a reflection of the plot of \(f(x)\) over the line \(y=x\). For \(x=0\), \(g(x)\) is undefined, and for \(x=1\), \(g(1)=0\). The function increases slowly as \(x\) gets bigger and approaches negative infinity as \(x\) moves towards 0 from the right.
04

Drawing the Graphs Together

Now, draw both graphs on the same coordinate plane. All the key points and features should be indicated.

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Most popular questions from this chapter

True or False? In Exercises 83 and \(84,\) determine whether the statement is true or false. Justify your answer. The graph of \(f(x)=\log _{6} x\) is a reflection of the graph of \(g(x)=6^{x}\) in the \(x\) -axis.

For how many integers between 1 and 20 can you approximate natural logarithms, given the values \(\ln 2 \approx 0.6931, \ln 3 \approx 1.0986,\) and \(\ln 5 \approx 1.6094 ?\) Approximate these logarithms (do not use a calculator )

Writing a Natural Logarithmic Equation In Exercises \(53-56,\) write the exponential equation in logarithmic form. $$e^{-0.9}=0.406 \ldots$

Function \(\quad\) Value $$\text { 58. } f(x)=3 \ln x \quad x=0.74$$

Rewrite each verbal statement as an equation. Then decide whether the statement is true or false. Justify your answer. The logarithm of the product of two numbers is equal to the sum of the logarithms of the numbers.
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