Question

Prove the following:

`log  125/147 = 3 - 3 log 2 - 2 log 7 - log 3`

Answer 1

Given: `log (125/147)`

To Prove: `log (125/147) = 3 - 3 log 2 - 2 log 7 - log 3`

Proof [Step-wise]:

1. Start with the left side:

`log (125/147) = log 125 - log 147`

2. Factor the arguments:

= log (5³) – log (3 × 7²)

3. Use power and product rules:

= 3 log 5 – (log 3 + 2 log 7)

= 3 log 5 – log 3 – 2 log 7

4. Express log 5 using log 10 and log 2 log means base 10, so log 10 = 1: 

`log 5 = log (10/2)`

= log 10 – log 2 

= 1 – log 2

5. Substitute into step 3:

= 3(1 – log 2) – log 3 – 2 log 7

= 3 – 3 log 2 – log 3 – 2 log 7

Therefore, `log (125/147) = 3 - 3 log 2 - 2 log 7 - log 3`, as required.