Register

CAT 2018 Set-1 | Question: 71

• retagged by
1,817 views

1 Answer

1 1 vote

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 $: \text {D}$

• edited by
Voting & Accepting an answer helps improve the community
← Previous · CAT 2018 Set-1 | Que...Next → · CAT 2018 Set-1 | Que...
← Previous in category · CAT 2018 Set-1 | Que...Next in category → · CAT 2018 Set-1 | Que...
Position:
Show:
Answer:

Related questions

2 2 votes
1 answers 1 answer
1.7k
1.7k views
go_editor asked Mar 19, 2020
1,700 views
CAT 2018 Set-1 | Question: 78
If $x$ is a positive quantity such that $2^x=3^{\log_52}$, then $x$ is equal to$1+\log_3\dfrac{5}{3}$$\log_58$$1+\log_5\dfrac{3}{5}$$\log_59$
2 2 votes
1 1 answer
1.7k
1.7k views
go_editor asked Mar 19, 2020
1,715 views
CAT 2018 Set-1 | Question: 82
$\log_{12}81=p$, then $3\bigg (\frac{4-p}{4+p}\bigg)$ is equal to $\log_416$$\log_68$$\log_616$$\log_28$
2 2 votes
1 1 answer
1.9k
1.9k views
go_editor asked Mar 19, 2020
1,851 views
CAT 2018 Set-1 | Question: 67
Let $x, y, z$ be three positive real numbers in a geometric progression such that $x < y < z$. If $5x$, $16y$, and $12z$ are in an arithmetic progression then the common ...
2 2 votes
1 1 answer
2.4k
2.4k views
go_editor asked Mar 19, 2020
2,376 views
CAT 2018 Set-1 | Question: 68
A tank is fitted with pipes, some filling it and the rest draining it. All filling pipes fill at the same rate, and all draining pipes drain at the same rate. The empty t...
3 3 votes
1 answers 1 answer
2.0k
2.0k views
go_editor asked Mar 19, 2020
1,971 views
CAT 2018 Set-1 | Question: 69
Given that $x^{2018}y^{2017}=1/2$ and $x^{2016}y^{2019}=8$, the value of $x^2+y^3$ is$35/4$$37/4$$31/4$$33/4$