Question

what is the inverse of $f(x)=2\ln(x-4)$?○ $f^{-1}(x)=e^{2x}-4$○ $f^{-1}(x)=e^{\frac{x}{2}}+4$○ $f^{-1}(x)=\frac{1}{2}e^{x}+4$○ $f^{-1}(x)=e^{\frac{x}{2}}-4$question 1010 ptswhat is the base of a common logarithm?○ $\pi$○ $e$○ 2○ 10

Explanation:

Step1: Set $y=f(x)$

$y = 2\ln(x-4)$

Step2: Isolate the logarithmic term

$\frac{y}{2} = \ln(x-4)$

Step3: Convert to exponential form

$e^{\frac{y}{2}} = x-4$

Step4: Solve for $x$

$x = e^{\frac{y}{2}} + 4$

Step5: Swap $x$ and $y$ for inverse

$f^{-1}(x) = e^{\frac{x}{2}} + 4$

A common logarithm is defined as the logarithm with a base of 10, denoted as $\log(x)$ (without a written base, it defaults to 10).

Answer:

For the inverse function: $\boldsymbol{f^{-1}(x) = e^{\frac{x}{2}} + 4}$ (matches the second option)
For the common logarithm base: $\boldsymbol{10}$ (matches the fourth option)