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Chapter 4: Problem 44

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

Short Answer

Expert verified
The graph of \(f(x) = 5^x\) will be a curve increasing rapidly for positive x-values and approaching \(y=0\) for negative x-values. The graph of \(g(x) = \log_{5} x\) will be a curve increasing at a decreasing rate for x-values larger than 0. The two functions intersect at the point (1,5).

Step by step solution

01

Graph the exponential function

Start by plotting \(f(x) = 5^x\), the exponential function. Even for negative x-values, the function will always be positive. For \(x=0\), \(y=1\) because any number to the power of 0 is 1. For \(x=1\), \(y=5\) because 5 to the power 1 equals 5. Therefore, the graph will increase rapidly for positive x-values and approach \(y=0\) for negative x-values, never reaching \(y=0\).
02

Graph the logarithmic function

Next, plot \(g(x) = \log_{5} x\), the logarithmic function. For this function, only positive x-values are allowed, as the logarithm of a negative number is not a real number. And the function equals 0 at \(x=1\) because the base 5 raised to the power 0 gives 1. The logarithm of 5 to the base 5 equals 1, because 5 to the power 1 equals 5. So like any logarithmic function, this graph will increase at a decreasing rate.
03

Identify intersection points

Finally, identify where the two graphs intersect. It should be at \(x=1, y=5\). The reason for this intersection is that exponential and logarithmic functions are inverse of each other.

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Most popular questions from this chapter

Evaluate or simplify each expression without using a calculator. $$ e^{\ln 300} $$

Explain why the logarithm of 1 with base \(b\) is \(0 .\)

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I estimate that \(\log _{8} 16\) lies between 1 and 2 because \(8^{1}=8\) and \(8^{2}=64\).

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Determine whether each statement makes sense or does not make sense, and explain your reasoning. I've noticed that exponential functions and logarithmic functions exhibit inverse, or opposite, behavior in many ways. For example, a vertical translation shifts an exponential function's horizontal asymptote and a horizontal translation shifts a logarithmic function's vertical asymptote.
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