Question

Prove the following:

2 log₁₀ 20 + log 5 – log 2 = 3

Answer 1

Given: 2 log₁₀ 20 + log 5 – log 2

To Prove: 2 log₁₀ 20 + log 5 – log 2 = 3

Proof [Step-wise]:

1. Use the power rule:

2 log₁₀ 20

= log₁₀ (20²)

= log₁₀ 400

2. Substitute into the expression:

2 log₁₀ 20 + log 5 – log 2

= log₁₀ 400 + log₁₀ 5 – log₁₀ 2

3. Use the product rule:

log₁₀ 400 + log₁₀ 5

= log₁₀ (400 × 5) 

= log₁₀ 2000

4. Use the quotient rule:

log₁₀ 2000 – log₁₀ 2

= `log_10 (2000/2)` 

= log₁₀ 1000

5. Recognize 1000 = 10³

So log₁₀ 1000

= log₁₀ (10³) 

= 3 log₁₀ 10

= 3 × 1

= 3

2 log₁₀ 20 + log 5 – log 2 = 3, as required.