Question

Given log 2 = a, log 3 = b, express the following in terms of a or b or both.

log 0.036

Answer 1

We are given:

• log 2 = a
• log 3 = b

We are to express:

log 0.036

Step 1: Express 0.036 as a fraction

`0.036 = 36/1000 = (2^2 * 3^2)/10^3`

Now recall:

10³ = 2³ · 5³

So:

`log 0.036 = log ((2^2 * 3^2)/(2^3 * 5^3)) = log (2^-1 * 3^2 * 5^-3)`

Step 2: Apply log rules

log 0.036 = – log 2 + 2 log 3 – 3 log 5 = – a + 2b – 3 log 5

log 0.036 = – a + 2b – 3 log 5