# Thread: Logarithm help need explanation

1. ## Logarithm help need explanation

Write 1 - 2log7x as a single logarithm.
Thanks!

2. ## Re: Logarithm help need explanation

Originally Posted by dzomberg
Write 1 - 2log7x as a single logarithm.
Thanks!

$\displaystyle \log_7(7)-\log_7(x^2)=\log_7\left(\frac{7}{x^2}\right)~.$

3. ## Re: Logarithm help need explanation

$\displaystyle $$1{\rm{ }} - {\rm{ }}2{\log _7}x = {\log _7}7 - {\log _7}{x^2} = {\log _7}{7 \over {{x^2}}}$$$

4. ## Re: Logarithm help need explanation

Hi, this kind of problems involves the use of properties. So you want that expression to be simplified in one logarithm.

First, let's write everything as a logarithm. We know that: $\displaystyle log_n (n) = 1$ so we could say that $\displaystyle n = 7$ (in our case). Thus $\displaystyle 1 = log_7 (7)$

Then, we have to put, for the second part, "inside" the logarithm. So we also know that: $\displaystyle a \cdot log_n(x) = log_n(x^a)$. Finally $\displaystyle 2 \cdot log_7(x) = log_7(x^2)$

We put it altogether (using the substraction of logarithm property of the same base) $\displaystyle log_n(x)-log_n(y) = log_n\left(\frac{x}{y}\right)$:

$\displaystyle 1 - 2\cdot log_7(x) = log_7(7) - log_7(x^2) = log_7\left(\frac{7}{x^2}\right)$