Question

If a^(3-x) × b^5x = a^(5+x) × b^3x , show that x log (b/a) = log a.​

Answer 1

Logarithmic problem

Given: $a^{3-x}\times b^{5x}=a^{5+x}\times b^{3x}$

To show: $x\:log(b/a)=log(a)$

Proof:

Given, $a^{3-x}\times b^{5x}=a^{5+x}\times b^{3x}$

$\Rightarrow a^{3-x}\times a^{-(5+x)}=b^{3x}\times b^{-5x}$

$\Rightarrow a^{3-x}\times a^{-5-x}=b^{3x}\times b^{-5x}$

$\Rightarrow a^{3-x-5-x}=b^{3x-5x}$

$\Rightarrow a^{-2x-2}=b^{-2x}$

$\Rightarrow a^{x+1}=b^{x}$

$\Rightarrow a^{x}\times a=b^{x}$

$\Rightarrow b^{x}/a^{x}=a$

$\Rightarrow (b/a)^{x}=a$

Taking $log$ to both sides, we get

$log(b/a)^{x}=log(a)$

$\Rightarrow x\:log(b/a)=log(a)$

This completes the proof.

Result: $x\:log(b/a)=log(a)$