Test Index

UPSC CDS I 2025 Mathematics Paper

© examsnet.com

Question : 20 of 100

Marks: +1, -0

If

\frac{log_{10}(100001-4^{x})}{5-x}=1

, then what is x equal

0
1
10
100

Solution:

Concept:

If

\frac{\log_{10} A}{B} = 1

, then

\log_{10} A = B

and

A = 10^{B}

.

Explanation:

Start with

\frac{\log_{10}(100001-4^{x})}{5-x} = 1

.

Multiply both sides by

(5-x)

:

\log_{10}(100001-4^{x}) = 5-x

.

Rewrite in exponential form:

100001-4^{x} = 10^{5-x}

.

Now test each option.

For

x=0

:

100001-4^{0}=100001-1=100000

and

10^{5-0}=10^{5}=100000

. Both sides equal. So

x=0

works.

For

x=1

: LHS =

100001-4=99997

, RHS =

10^{4}=10000

, not equal.

For

x=10

: LHS =

100001-4^{10}=100001-1048576=-948575

, RHS =

10^{-5}=0.00001

, not equal.

For

x=100

:

4^{100}

is huge, LHS negative, RHS

10^{-95}

tiny positive, not equal.

Thus only

x=0

satisfies.

Answer:

x = 0

(Option A).

© examsnet.com

Go to Question:

Prev Question

Next Question