Question

Prove that a to the power log b base x= b to the power log a base x ​

Answer 1

Answer:

( logₓ b ) ( logₓ a ) = ( logₓ a ) ( logₓ b )         ... (*)

Now use the fact that

p logₓ q = logₓ (q^p).

For the LHS of (*), use p = logₓ b and q = logₓ a.

For the RHS of (*), use p = logₓ a and q = logₓ b.

This gives:

$\log_x a^{\log_xb} = \log_x b^{\log_xa}$

Now raising x to the power of each of these (i.e. getting rid of the logₓ from each side), we are left with the required result:

$a^{\log_xb} = b^{\log_xa}$