What is the solution of the equation x {\log}_{10}{10/3}+\log_{10}3={\log}_{10}(2+3^x)+x? [2017 CDS-I]

UPSC 2017 CDS I Math Paper

Question : 16 of 100

Marks: +1, -0

What is the solution of the equation

x‌log10

+log103=log10(2+3x)+x?

[2017 CDS-I]

Solution:

xlog10(‌‌

)+log103
=log10(2+3x)+x
⇒x[log10(‌‌

)−1]+log103
=log10(2+3x)

⇒x[log10(‌‌

)−log1010]+log103

=log10(2+3x)
⇒xlog10[‌‌

(‌‌

)

]+log103
=log10(2+3x)
⇒xlog10(‌‌

)+log103
=log10(2+3x)
⇒log10(‌‌

)=log10(2+3x)
⇒‌‌

=2+3x
For x=0,
LHS=

RHS=2+3x=2+30=2+1=3>Thus, the solution of the given equation is x = 0.

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