I am a little confused about composition of logarithms, I am given $f(x)=5^{2+5x}$ and $g(x)=\log_5 x$ and I am supposed to find $f(g(x))$. Here is what I have done so far,
$$5^{2+5(\log_5 x)}$$
$$5^{2+5\log_x 5}$$
I am stuck on this step because I don't know how to cancel the log in the exponent. Since it's $5\log_5 x$ and not $5^{\log_5 x}$. Could anyone explain how I would simplify this problem?
Best Answer
If you recall, there are several important properties of logarithm. One of them, which your class should have mentioned, is that $p \log_b{x} = \log_b(x^p)$ for any choices of $x > 0$, real number $p$, and base $b$. This should allow you to simplify your problem in the following way:
$$5^{2 + 5 \log_5{x}} = 5^{2 + \log_5{x^5}}$$ Now, remember that whenever we have $x^{a + b}$, we can break it up into $x^{a+b} = (x^a)(x^b)$.