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Chapter 3: Problem 71

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 \(g(e^{5}) = 5\).

Step by step solution

01

Analyze the given function

The function given is \(g(x)=\ln x\). This is a logarithmic function with base \(e\). The value we need to find is \(g(e^{5})\) which means we need to substitute \(x=e^{5}\) in the function.
02

Substitute the given value into the function

Replace \(x\) with \(e^{5}\) in the function \(g(x)\). So, the function becomes \(g(e^{5})=\ln(e^{5})\).
03

Apply logarithmic properties

As the base of the logarithm and the base of the exponent are the same, we can simply use the rule of logarithms that states \(\ln a^{b} = b \cdot \ln a\). However, since \(\ln e = 1\), this simplifies further to \(\ln e^{b} = b\). Thus, \(\ln(e^{5}) = 5\).

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