How am I supposed to simplify this logarithm?

Question

Given is $$\log_6(a)=6$$

Simplify - $$\log_6 (1/a^7)$$ and the answer is supposed to be $-42$.

I don't understand what I'm supposed to do here?

Answer 1

HINT

Recall that by definition

$$\log_6a=6 \iff 6^6=a$$

then

$$\log_6\left(\frac1{a^7}\right)=x \iff 6^x=\frac1{a^7}$$

or use that

• $\log x^n=n\log x$
• $\log \frac1x=-\log x$

Answer 2

For any logarithm, $\log a^b=b \log a$, so in your case $$\log_6\frac 1{a^7}=-7\log_6a=-42$$ We don't have to solve for $a$ at all, though we could use $\log_6a=6$ to say $a=6^6$