91Ó°ÊÓ

We are given the following functions: \[f(x) = \log_{2x}7\] \[g(x) = 10^x\] Our task is to compute the composition function \((f \circ g)(x)\).

The composition function is defined as \((f \circ g)(x) = f(g(x))\). So, in our case, we need to find \(f(g(x))\). Start by computing the expression for \(g(x)\): \[g(x) = 10^x\] Now, we need to evaluate \(f(g(x))\). To do this, substitute \(g(x)\) into \(f(x)\): \[f(g(x)) = \log_{2(10^x)}7\]

Now, let's simplify the expression for \(f(g(x))\). Notice that the base of the logarithm is \(2(10^x)\). We can simplify this by writing it as a product: \[f(g(x)) = \log_{2(10^x)}7 = \log_{2 \cdot 10^x}7\] Now, we'll use the logarithm property that states that \(\log_{a \cdot b}{x} = \frac{\log_{a}{x}}{\log_{b}{x}}\). Apply this property to our expression: \[f(g(x)) = \frac{\log_{2}{7}}{\log_{10^x}{7}}\] Finally, we use another logarithm property that states \(\log_{a^b}{x} = \frac{\log_a{x}}{b}\). Applying this property, we get: \[f(g(x)) = \frac{\log_{2}{7}}{\frac{\log_{10}{7}}{x}}\] To simplify further, we can multiply both the numerator and denominator by \(x\): \[f(g(x)) = \frac{x \cdot \log_{2}{7}}{\log_{10}{7}}\] Now, we have the formula for the composition function \((f \circ g)(x)\): \[(f \circ g)(x) = \frac{x \cdot \log_{2}7}{\log_{10}7}\]