Question

Evaluate the following: 

`log 100 + 2 log 0.01 - 1/2 log 5^-4 + 2/3 log 8`

Answer 1

Given: `log 100 + 2 log 0.01 - 1/2 log 5^-4 + 2/3 log 8`

Step-wise calculation:

1. log 100

= log (10²)

= 2

2. 2 log 0.01

= 2 × log (10^–2) 

= 2 × (–2) 

= –4

3. `-1/2 log 5^-4`

= `(-1/2) xx (-4 log 5)` 

= 2 log 5

4. `2/3 log 8`

= `(2/3) xx log (2^3)` 

= `(2/3) xx (3 log 2)` 

= 2 log 2

5. Combine:

2 + (–4) + 2 log 5 + 2 log 2 

= –2 + 2(log 5 + log 2)

6. log 5 + log 2

= log (5 × 2) 

= log 10

= 1

So 2(log 5 + log 2)

= 2 × 1 

= 2

7. Therefore,

−2 + 2

= 0