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Chapter 3: Problem 74

Evaluate \(g(x)=\ln x\) at the indicated value of \(x\) without using a calculator. $$x=e^{-5 / 2}$$

Short Answer

Expert verified
The value of the function \(g(e^{-5 / 2})\) is -2.5.

Step by step solution

01

Substitute x in the Function

The first step is to substitute \(x = e^{-5 / 2}\) into the function \(g(x)\). Our function becomes: \(g(e^{-5 / 2}) = \ln e^{-5 / 2}\)
02

Use the Logarithm Property

The property of the natural logarithm function says that: \(\ln(a^b) = b*\ln(a)\). Here the base \(a\) is \(e\) and the exponent \(b\) is \(-{5 / 2}\). So, applying this property to our equation: \(\ln(e^{-5 / 2}) = -5 / 2 * \ln(e)\)
03

Solve the Function

\(\ln(e)\) simplifies to 1 since the natural logarithm function is the inverse of the exponential function. So our equation simplifies to: \(-5 / 2 * 1 = -2.5\). So, \(g(e^{-5 / 2}) = -2.5\)

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Most popular questions from this chapter

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