How to express $\log_3(2^x)$ using $\log_{10}$? And how to evaluate $4^{\log_4y}$?

Question

How to express $\log_3(2^x)$ using $\log_{10}$? And how to evaluate $4^{\log_4y}$?

Answer 1

Hint 1: $$\log_a b = \frac{\log_{c} b}{\log_{c} a}.$$

Hint 2: $$a^{\log_a b} = b.$$

Answer 2

Generally without accuracy, $\log_a(b)=\dfrac{\log_c(b)}{\log_c(a)}$ and $a^{\log_a(b)}=b$, where a, b, c are positive numbers. Hope this can help you.

Answer 3

The second is clear from the definition of $\log_4(y)$.

The first follows from the formula $\log_a(x) = \frac{\log_b(x)}{\log_b(a)}$plus the fact we can pull exponents out of the $\log$.

(Know the basic properties!)