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Chapter 3: Problem 61

Evaluate \(g(x)=\ln x\) at the indicated value of \(x\) without using a calculator. $$x=e^{5}$$

Short Answer

Expert verified
The value of \(g(e^{5})\) is 5.

Step by step solution

01

Understanding the function

Firstly, it must be understood that the given function: \(g(x) = ln(x)\) is a logarithmic function using base \(e\). This function takes any positive number as its input and returns a real number as output.
02

Plugging in the given value of x into the function

You are required to evaluate the function at \(x = e^{5}\). This process simply involves replacing the \(x\) in \(g(x)\) with \(e^{5}\). After this replacement, the function becomes: \(g(e^{5}) = ln(e^{5})\).
03

Evaluating the function

To evaluate the function at \(x = e^{5}\), substitute \(e^{5}\) into the function. Using the fact that the natural logarithm is the inverse function of the exponential function with base \(e\), we have that \(ln(e^{n}) = n\) for any number \(n\). From this, we can deduce that \(ln(e^{5}) = 5\). Therefore, \(g(e^{5}) = 5\).

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Most popular questions from this chapter

Writing a Natural Logarithmic Equation In Exercises \(53-56,\) write the exponential equation in logarithmic form. $$e^{1 / 2}=1.6487 \ldots$$

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