If $\log _{2}\left(5+\log _{3} a\right)=3$ and $\log _{5}\left(4 a+12+\log _{2} b\right)=3$, then $a+b$ is equal to

We have two logarithmic equations to solve:

$\log_2(5 + \log_3 a) = 3$

$\log_5(4a + 12 + \log_2 b) = 3$

When we see $\log_x y = z$, it means $x^z = y$. This is the fundamental relationship we'll use throughout.

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Starting with: $\log_2(5 + \log_3 a) = 3$

We're asking "2 raised to what power equals $(5 + \log_3 a)$?"

Since the answer is 3, we have:

$2^3 = 5 + \log_3 a$

$8 = 5 + \log_3 a$

$\log_3 a = 3$

Now we have another logarithm to solve: $\log_3 a = 3$ means "3 raised to what power equals $a$?"

$a = 3^3 = 27$

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Starting with: $\log_5(4a + 12 + \log_2 b) = 3$

We're asking "5 raised to what power equals $(4a + 12 + \log_2 b)$?"

Since the answer is 3:

$5^3 = 4a + 12 + \log_2 b$

$125 = 4a + 12 + \log_2 b$

Substituting $a = 27$:

$125 = 4(27) + 12 + \log_2 b$

$125 = 108 + 12 + \log_2 b$

$125 = 120 + \log_2 b$

$\log_2 b = 5$

Solving this logarithm: $\log_2 b = 5$ means "2 raised to what power equals $b$?"

$b = 2^5 = 32$

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We now have:

$a = 27$

$b = 32$

Therefore:

$a + b = 27 + 32 = 59$