Question

prove that 3^log 4= 4^log 3

Answer 1

Given, $\mathbf{3^{log4}=4^{log3}}$
LHS =y = $\mathbf{3^{log4}}$
Take log both sides,
logy = log4.log3 [ as you know, loga^n = nloga , so, log3^{log4}=log4.log3]
logy = log3.log4 = log4^{log3}
Logy = log4^{log3}
Remove log from both sides,
y = 4^{log3} = RHS

Hence proved

Answer 2

Answer and explanation:

To prove : $\mathbf{3^{\log4}=4^{\log3}}$

Proof :

Taking LHS,

$y={3^{\log4}}$

Taking log both sides,

$\log y=\log {3^{\log4}}$

We know, $\log a^n = n\log a$

$\log y=\log4\log {3}$

Apply commutative, $ab=ba$

$\log y=\log3\log {4}$

Write it as,

$\log y=\log {4}^{\log3}$

Remove log from both sides,

$y={4}^{\log3}$

= RHS

Hence proved.