How do you solve this using only given values, logarithm rules and no calculator?

Question

Given that $\log12=1.0792$ and $\log4=0.6021$, solve $\log8$ without a calculator.

I am familiar with the following three rules:

1. Product rule: $\log(a\cdot b)=\log a+\log b$
2. Quotient rule: $\log(a/b)=\log a-\log b$
3. Power rule: $\log(a^b)=b\cdot\log a$

But I honestly don't see how they help in this case. Any way you slice it, it seems like it's necessary to introduce $\log2$, which is not given. Am I missing something?

Answer 1

$\log 4 = \log 2^2 =2\log 2 =0.6021$
$\implies\log 2 =0.3010$
$\log 8 = \log 2^3 = 3\log 2 = 3*0.3010 = 0.9030$

Answer 2

Hint: $8^2=4^3$.

Alternative hint: $2=\sqrt4\,$.

Answer 3

$\log 8=\log(4^{3/2})$

Use the $3^{\text{rd}}$ rule.

Answer 4

Since you received the answers, let me be slightly more general and show you what you can do just knowing that $\log_{10}(12)=1.0792$ and $\log_{10}(4)=0.6021$. Since $4=2^2$, you then have $\log_{10}(2)=0.3010$. Since $3=\frac{12} {4}$, you then have $\log_{10}(3)=0.4771$. We could continue like that for quite many numbers.

But, by the end, wht this means is that, just based on this mimited information, you can compute the logarithm of any number which write $2^a3^b4^c6^d8^e9^f10^g$.