Question

Prove that: (1 + log_a b).log_ab x = log_a x.

Answer 1

Given:

a > 0,

a ≠ 1, 

b > 0,

ab > 0 and ab ≠ 1,

x > 0 so all logarithms below are defined.

To Prove: (1 + log_a b) × log_ab x = log_a x

Proof [Step-wise]:

1. Start with the change-of-base formula for logarithms:

`log_(ab) x = (log_a x)/(log_a (ab))`

2. Evaluate log_a (ab) using log rules:

log_a (ab) 

= log_a a + log_a b 

= 1 + log_a b

3. Substitute step 2 into step 1:

`log_(ab) x = (log_a x)/(1 + log_a b)`

4. Multiply both sides of this equality by 1 + log_a b: 

`(1 + log_a b) xx log_(ab) x = log_a x`

Hence (1 + log_a b) × log_ab x = log_a x, as required.