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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?

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up vote 3 down vote accepted

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 $x^{a+b} = (x^a)(x^b)$.

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I tried moving the 5 to the exponent for $log_{5}x$ but I am still stuck. What am I supposed to do now? – Kot Oct 5 '12 at 0:35
@StevenN, I've elaborated a little bit more above. – Christopher A. Wong Oct 5 '12 at 0:39
Wow, I forgot about that property. Thank you for reminding me of it :). I understand how to solve this now. – Kot Oct 5 '12 at 0:47

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