Logarithm problem when simplifying without answering in decimals

Question

I'm trying so solve the following problem:

$$\log(8) + \log(x) = 6 \, \log(5).$$

The answer is supposed to be "exact" so without any decimals and as simplified as possible.

However, the $6$ in $6 \log(5)$ gives me a hard time to do this. Can anyone show me the way to solve this?

Answer 1

We use the following rules of logarithms here:

1. $\log(a)+\log(b)=\log(ab)$
2. $c\log(d)=\log(d^c)$

With these rules in mind, our problem reduces to $$\log(8x)=\log(5^6).$$ Now, we raise both sides to the base $e$, known as exponentiation (the inverse of logarithms) and we get $$8x=5^6,$$ from which, we use trivial algebra to get $$x=\frac{5^6}{8}.$$

Answer 2

Hint

$$\log 8 +\log x= \log 8x $$

$$6\log 5=\log 5^6.$$