Question

Helppp pls
lesson is Logarithms and Logarithmic Functions

Answer 1

`f(g(x))=x`

`g(f(x))=x`

Explanation:

Given functions:

`f(x)=\log_(10)(x)`

`g(x)=10^x`

The composite function f[g(x)] means to substitute the function g(x) in place of the x in function f(x):

`\begin{aligned}f[g(x)]&=\log_(10)[g(x)]\\&=\log_(10)\left(10^x\right)\\&=x\log_(10)\left(10\right)\\&=x(1)\\&=x\end{aligned}`

The composite function g[f(x)] means to substitute the function f(x) in place of the x in function g(x):

`\begin{aligned}g[f(x)]&=10^(f(x))\\&=10^{\log_(10)(x)}\\&=x\end{aligned}`

Log rules used

`\log_ax^n=n\log_ax`

`\log_aa=1`

`\:a^(\log _a\left(x\right))=x`