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Chapter 3: Problem 74

Find a formula for \((f \circ g)(x)\) assuming that \(f\) and \(g\) are the indicated functions. \(f(x)=\log _{2 x} 7\) and \(g(x)=100^{x}\)

Short Answer

Expert verified
The composite function \((f \circ g)(x) = \log_{2(100^x)}7\).

Step by step solution

01

Recall the Definition of the Composite Function

Let's recall the definition of the composite function \((f \circ g)(x) = f(g(x))\). We need to substitute the function \(g(x)\) into the function \(f(x)\) to obtain the composite function.
02

Substitute \(g(x)\) into \(f(x)\)

Now, substitute \(g(x)=100^x\) into \(f(x)=\log_{2x}7\): \((f \circ g)(x) =f(g(x))=f(100^x)\)
03

Replace the variable \(x\) with \(100^x\) in \(f(x)\)

Replace the variable \(x\) in the function \(f(x)\) with the expression \(100^x\), \((f \circ g)(x) = \log_{2(100^x)}7\)
04

Simplify the expression

The function is already in its simplest form: \((f \circ g)(x) = \log_{2(100^x)}7\) Now we have the composite function \((f \circ g)(x) = \log_{2(100^x)}7\).

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