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Chapter 3: Problem 7

Express the function \(f(x)=g(x)^{h(x)}\) in terms of the natural logarithmic and natural exponential functions (base \(e\) ).

Short Answer

Expert verified
Question: Express the function \(f(x) = g(x)^{h(x)}\) in terms of natural logarithmic and natural exponential functions. Answer: \(f(x) = e^{h(x) \cdot \ln(g(x))}\)

Step by step solution

01

Apply the natural logarithm on both sides of the equation

To perform this step, we need to take the ln of both sides of the given function. This will help us to separate the terms and apply the properties of logarithms: \[\ln(f(x)) = \ln(g(x)^{h(x)})\]
02

Use the logarithm power rule

The power rule states that \(\ln(a^b) = b\cdot \ln(a)\). We can apply this rule to the right side of the equation to simplify the expression: \[\ln(f(x)) = h(x) \cdot \ln(g(x))\]
03

Isolate the function \(f(x)\)

Our main goal is to rewrite the function \(f(x)\) in terms of natural logarithmic and natural exponential functions, so we need to express \(f(x)\) on one side of the equation. To do this, we can use the inverse of natural logarithm, which is a natural exponential function: \[f(x) = e^{\ln(f(x))}\] Now the equation becomes: \[e^{\ln(f(x))} = e^{h(x) \cdot \ln(g(x))}\]
04

Simplify the expression

Since the natural exponential and natural logarithm functions are inverses of each other, the left side of the equation simplifies to just \(f(x)\): \[f(x) = e^{h(x) \cdot \ln(g(x))}\] Now the function \(f(x)\) is expressed in terms of natural logarithmic and natural exponential functions.

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