I'm stuck on solving what my textbook calls an "advanced logarithm problem". Basically, it's a logarithmic equation with logarithms of different bases on either side. My exercise looks like this:
$$2 \log_2 x – \log_2 (x – \tfrac1 2) = \log_3 3$$
To start off, I used the power rule to simplify the first term to get this:
$$\log_2 x^2 – \log_2 (x – \tfrac1 2) = \log_3 3$$
Then I used the quotient rule to get this:
$$\frac {x^2} {x – \frac 1 2} = \log_3 3$$
Then I turned the logarithmic equation into an exponential equation to get this:
$$3^{\frac {x^2} {x – \frac 1 2}} = 3$$
Now, however, I'm unsure of how to proceed. The textbook has neither explained to me how to simplify such complex exponents nor do such exponents have any precedence. I'm therefore assuming that I went wrong somewhere previously in solving the problem, but as far as I can tell I did everything by the book.
Did I go wrong? And if not, am I really supposed to simplify that exponent?
Best Answer
Sorry, read too quickly at first. You were on the right track - but you lost a logarithm at this step: $$\frac{x^2}{x-\frac{1}{2}}=\log_3(3)$$ It should be $$\log_2\left(\frac{x^2}{x-\frac{1}{2}}\right)=\log_3(3)$$ Now, what is $\log_3(3)$?