TITLE: Algebra with exponential functions
QUESTION [3 upvotes]: If $f(x) = 4^x$ then show the value of $$f(x+1) - f(x)$$ in terms of $f(x)$.
I know the answer is $3f(x)$ because
$f(x+1)$ means that it is $4^x$ multiplied by 4 once more, which minus one is 3.
The question: How do I show this process algebraically? (Hints only please) I have tried using ln() functions to remove the powers to no avail.
$$\ln(f(x)) = x\ln(4)$$
$$\ln(f(x+1)) = (x+1)\ln(4) = x\ln(4) + \ln(4)$$
$$\ln(f(x+1)) = \ln(f(x)) + \ln(4)$$
from here I don't know how to remove the natural logs to replace $f(x+1)$. What is a different approach I should use?

REPLY [1 votes]: Let $y = f(x)$. You want $f(x+1) - f(x)$ to be in terms of $y$, so you need to replace $x$ with an expression of $y$.
This expression is the inverse function of $f(x)$: $$x=f^{-1}(y)=\log_4y$$
Therefore, $$f(x+1) - f(x)=f(f^{-1}(y)+1)-f(f^{-1}(y))$$
Simplify this expression to get $3y$.