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If $$5^{a} = 9^{b} = 2045$$, then the value of $$\frac{ab}{a + b}$$ = _____.
It is given,
$$5^a=9^b=2045$$
$$5^a=2045$$
Applying log on both the sides, we get
$$\log_55^a=\log_52045$$
$$a=\log_52045$$
Similarly, we get $$b=\log_92045$$
$$\dfrac{ab}{a+b}=\dfrac{1}{\dfrac{1}{a}+\dfrac{1}{b}}$$
$$\dfrac{1}{a}=\log_{2045}5$$
$$\dfrac{1}{b}=\log_{2045}9$$
$$\dfrac{1}{a}+\dfrac{1}{b}=\log_{2045}45$$
$$\dfrac{ab}{a+b}=\log_{45}2045\ =\ 2$$
The answer is option C.
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