Question

Express as single log:

`1/3 log 125 - 2 log 3 + 1/4 log 625`

Answer 1

`1/3 log 125 - 2 log 3 + 1/4 log 625`

Step 1: Apply the logarithmic power rule:

`log (125)^(1/3) - log (3)^2 + log (625)^(1/4)`

Step 2: Simplify powers:

`125^(1/3)` = 5   ...(Since 125 = 5³)

`625^(1/4)` = 5   ...(Since 625 = 5⁴)

So the expression becomes:

`log 5 - log 9 + log 5`

Step 3: Combine using log rules:

⇒ log 5 + log 5

= log (5 × 5)

= log 25

⇒ log 25 − log 9

= `log (25/9)`