TITLE: Proving $x$ is a given quotient of logarithms
QUESTION [5 upvotes]: I'm practicing some questions on logarithms at the moment in order that I'm up to speed with the problem solving aspect before I embark on my PHD in chemical engineering at Boston college next year.
I've been studying the laws of logarithms and what I am to do when it is necessary to add and subtract logs. So for example, for the first part of the question that is causing me trouble, I would utilize the first law of logarithms.
My issue comes with the "Prove part", and ascertaining a value for $x$ when it is already involved in the first part of the question.
I know I'll most likely to be shut down for lack of evidence and research for this, but I would like to know what the most logical first step would be and which rule I need to follow. It doesn't follow on from the other questions I've been solving and so I can't problem solve it as easily as I can the rest.
Here is my question:
If $4^x\cdot 5^{3x+1}=10^{2x+1}$, prove that $x=\dfrac{\log(2)}{\log(5)}$.
Thanks, not a problem if I get shut down, I know this isn't in the spirit of the website and would normally never ask a question in such a manner.

REPLY [6 votes]: Alternative Solution:
Recall the laws of logarithms:

$\log a^b = b \log a$.
$\log ab = \log a + \log b$
$\log \frac{a}{b} = \log a - \log b$

By taking the logarithm on both sides of your equation and applying rule (1), you will get:
$$
x \log 4 + (3x + 1) \log 5 = (2x + 1) \log (10)
$$
which can be simplified to
$$
(\log 4 + 3 \log 5 – 2 \log 10) x = \log(10) - \log(5)
$$
Using the mentioned rules, we get
$$
\log\left(\frac{4 \times 5^3}{10^2}\right) x = \log\left(\frac{10}{5}\right)
$$
which establish your result.