Replace $f(x)$ with $y$ so you have $y=5(2)^{3x-4}$, then switch $x$ and $y$ to get an equation involving $x$ and $y$ that you can solve for $y$.

After swapping, first isolate the power of $2$ by dividing by $5$. Then use a logarithm with base $2$ to bring the exponent down, and finally solve the resulting linear equation for $y$.

Once you have an expression for $y$ in terms of $x$, rewrite it neatly and compare it to options A–D to see which one is exactly the same.

In Desmos, type f(x)=5*2^(3x-4) to graph the given function $f$.

Add the line y=x. The graph of the true inverse of $f$ will be the reflection of $f$ across this line.

Enter each option as a separate function, for example gA(x)=(log_2(5x)-4)/3, gB(x)=3*log_2(x/5)+4, gC(x)=(log_2(x/5)+4)/3, and gD(x)=(log_2(x)-4)/15. (Use log_2(...) for base-2 logs.)

Look for the candidate whose graph is exactly the mirror image of $f(x)$ across the line $y=x$. That candidate defines the correct inverse function $g(x)$.

Start by writing the function with $y$ instead of $x$:

For the inverse, we swap the roles of $x$ and $y$ and then solve for $y$:

We want to isolate $2^{3y-4}$.

Divide both sides of

by $5$:

Now the exponential term $2^{3y-4}$ is alone on one side.

Because the base of the exponential is $2$, take $\log_{2}$ of both sides:

Solve for $y$:

Therefore, the inverse is $g(x)=\dfrac{\log_{2}\!\left(\dfrac{x}{5}\right)+4}{3}$.