Question

Sometimes to solve an equation, we may use the identity $a^{{\log }_{a}b}=b,b>0,a>0,a\ne 1$

Then solution set of $3x^{{\log }_{5}2}+2^{{\log }_{5}x}=64$ is,

• A. $5$
• B. $25$
• C. $125$
• D. $625$

Answer 1

The correct option is D $625$

$3x^{{\log }_{5}2}+2^{{\log }_{5}x}=64$

Since, $a^{{\log }_{b}c}=c^{{\log }_{b}a}$

Therefore, $3\left(2^{{\log }_{5}x}\right)+2^{{\log }_{5}x}=64$

$\Rightarrow 4\cdot 2^{{\log }_{5}x}=64$

$\Rightarrow 2^{{\log }_{5}x}=16=2^4$

Equating power of 2

$\Rightarrow {\log }_{5}x=4$

$\Rightarrow x=5^4=625$

Ans: D