Solving a problem involving $\log$ function

Question

If $$a = \log_23 , b = \log_52$$ then what is $\log45$ ?

(I have to define $\log45$ using $a$ and $b$)

What I did : $$\log45 = 2\log3 + \log5$$ $$\log45 = \log2\left(2a + \frac1{b}\right)$$

Stuck here. Please help me

Answer 1

You can proceed as follows:

$$ \log 2 = \frac{1}{\log _2^{10}} = \frac{1}{\log_2^2 + \log_2^5} = \frac{1}{1+\frac{1}{b}}$$.

Answer 2

Try using these two identities:

$$a \log b = \log{b^a}$$

$$\log_b a = \frac{\ln a}{\ln b}$$