Taking exponents of two numbers to make them equal

Question

If $3^a = 5^b = m$ and $\frac{1}{a} + \frac{1}{b} = 2$, what is the value for m?

Since $a, b \neq 0$, I assumed both a and b are fractions, so I did:

Let $a = \frac{a_1}{a_2}; b = \frac{b_1}{b_2}$

Then we subsitute them in and get $3^{a_1 b_2} = 5^{a_2 b_1} = m$

Now we're stuck into a similar situation as our original equation. How can I move further?

Answer 1

Observe that we have $m>0.$

We get $a \ln 3 = \ln m$ and $b \ln 5= \ln m.$ This gives

$$2= \frac{\ln 3+ \ln5}{\ln m}= \frac{\ln 15}{\ln m}.$$

Can you take it from here ?

Answer 2

Hint: $m^{\frac 1a}=3$ and $m^{\frac 1b}=5$.