Question

Find the value of x in the following:

log x = –2 + 3 log 2 – 5 log 3 + 2 log 72 + log 3

Answer 1

Given: log x = –2 + 3 log 2 – 5 log 3 + 2 log 72 + log 3

Step-wise calculation:

1. Convert constants and use log rules log a + log b = log (ab); k log a = log (a^k): 

log x = log (10^–2) + log (2³) + log (72²) + log (3) + log (3^–5)

2. Combine into a single log:

log x = log (10^–2 × 2³ × 72² × 3 × 3^–5)

3. Prime-factor 72 = 2³ × 3², 

So 72² = 2⁶ × 3⁴. 

Collect exponents:

For 2: exponent

= (–2) + 3 + 6

= 7

⇒ 2⁷

For 3: exponent

= 4 + 1 + (–5)

= 0

⇒ 3⁰ = 1

For 5: exponent

= (–2) from 10^–2

⇒ 5^–2

Product

= 2⁷ × 5^–2 

= `128/25`

4. Thus, `log x = log (128/25)`

⇒ `x = 128/25`

⇒ x = 5.12