Question 19

If $$5^{a} = 9^{b} = 2025$$, then the value of $$\frac{ab}{a + b}$$ = _____.

Solution

It is given,

$$5^a=9^b=2025$$

$$5^a=2025$$

Applying log on both the sides, we get

$$\log_55^a=\log_52025$$

$$a=\log_52025$$

Similarly, we get $$b=\log_92025$$

$$\dfrac{ab}{a+b}=\dfrac{1}{\dfrac{1}{a}+\dfrac{1}{b}}$$

$$\dfrac{1}{a}=\log_{2025}5$$

$$\dfrac{1}{b}=\log_{2025}9$$

$$\dfrac{1}{a}+\dfrac{1}{b}=\log_{2025}45$$

$$\dfrac{ab}{a+b}=\log_{45}2025\ =\ 2$$

The answer is option C.

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