# CAT 2018 Set-1 | Question: 71

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If $\log_2(5+\log_3a)=3$ and $\log_5(4a+12+\log_2b)=3$, then $a+b$ is equal to

1. $67$
2. $40$
3. $32$
4. $59$

Given that,

• $\log_{2} (5+ \log_{3} {a}) = 3 \quad \longrightarrow (1)$
• $\log_{5} ( 4a + 12) = 3 \quad \longrightarrow (2)$

From equation $(1),$

$\log_{2} (5 + \log_{3}{a}) = 3$

$\Rightarrow 5 + \log_{3}{a} = 2^{3}$ $\quad [ \because {\log_{a}{b}} = x \Rightarrow b = a^{x}]$

$\Rightarrow \log_{3}{a} = 3$

$\Rightarrow a = 3^{3}$

$\Rightarrow \boxed {a = 27}$

From equation $(2),$

$\log_{5} (4a + 12 + \log_{2}{b}) = 3$

$\Rightarrow 4a + 12 +\log_{2}{b} = 5^{3}$

$\Rightarrow 4(27) + 12 + \log_{2}{b} = 125$

$\Rightarrow 108 + 12 + \log_{2}{b} = 125$

$\Rightarrow 120 + \log_{2}{b} = 125$

$\Rightarrow \log_{2}{b} = 125 – 120$

$\Rightarrow \log_{2}{b} = 5$

$\Rightarrow b = 2^{5}$

$\Rightarrow \boxed {b = 32}$

$\therefore$ The value of $a+b = 27 + 32 = 59$

Correct Answer $: \text {D}$

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