TheCATExam
Sign in

CAT 2018 Slot 1 — QA Question 24

LogarithmsEasy

If log2(5 + log3 a) = 3 and log5(4a + 12 + log2 b) = 3, then a + b is equal to

Answer & solution

Correct answer: 59

  • A

    40

  • B

    67

  • 59

  • D

    32

Solution

Easy

Peel the logarithms off one at a time. Convert each outer log equation into exponential form to expose the inner expression, solve for aa, then feed that into the second equation to solve for bb.

1

Unwrap the first equation for aa. Convert log⁡2( ⋅ )=3\log_2(\,\cdot\,)=3 to base-2 exponential form.

log⁡2 ⁣(5+log⁡3a)=3⇒ 5+log⁡3a=23=8(exponential form)⇒ log⁡3a=3⇒ a=33=27\begin{aligned} &\log_2\!\left(5+\log_3 a\right)=3\\ &\Rightarrow\ 5+\log_3 a = 2^3 = 8 \quad\text{(exponential form)}\\ &\Rightarrow\ \log_3 a = 3\\ &\Rightarrow\ a = 3^3 = 27 \end{aligned}
2

Substitute a=27a=27 into the second equation. Convert log⁡5( ⋅ )=3\log_5(\,\cdot\,)=3 to base-5 exponential form.

log⁡5 ⁣(4a+12+log⁡2b)=3⇒ log⁡5 ⁣(4(27)+12+log⁡2b)=3(from step 1)⇒ 108+12+log⁡2b=53=125⇒ log⁡2b=125−120=5⇒ b=25=32\begin{aligned} &\log_5\!\left(4a+12+\log_2 b\right)=3\\ &\Rightarrow\ \log_5\!\left(4(27)+12+\log_2 b\right)=3 \quad\text{(from step 1)}\\ &\Rightarrow\ 108+12+\log_2 b = 5^3 = 125\\ &\Rightarrow\ \log_2 b = 125-120 = 5\\ &\Rightarrow\ b = 2^5 = 32 \end{aligned}
3

Add the two values.

a+b=27+32=59a+b = 27+32 = 59
a+b=59a+b = 59

Related Logarithms questions

See all Logarithms questions →
CAT 2018 Slot 1 QA Q24: If log 2 (5 + log 3 a) = 3 and log 5 (4a + 12 + log 2 b) = 3, then a + b is equal to — Solution | TheCATExam