1. Logarithms – Learning Outcomes
Solve problems using the laws for logarithms:
log 𝑎 (𝑥𝑦)
log 𝑎 𝑥 𝑦
log 𝑎 𝑥 𝑞
log 𝑎 𝑎
log 𝑎 1
log 𝑎 𝑥 = log 𝑏 𝑥 log 𝑏 𝑎

2. Solve Problems Using the Laws for Logs
Logarithms are the inverse operation of exponentiation.
Notation: log 𝑎 𝑐=𝑏 ⇔ 𝑎 𝑏 =𝑐
Alternatively, consider the form log 𝑎 𝑎 𝑏 =𝑏
Watch out for common logs - log 10 and log 𝑒 = ln

3. Solve Problems Using log 𝑎 (𝑥𝑦)
Given that log 𝑎 𝑐=𝑏 , and recalling 𝑎 𝑝 × 𝑎 𝑞 = 𝑎 𝑝+𝑞 , prove that log 𝑎 𝑥𝑦 = log 𝑎 𝑥 + log 𝑎 𝑦 .
To make it easier, let log 𝑎 𝑥 =𝑝 and log 𝑎 𝑦 =𝑞

4. Solve Problems Using log 𝑎 𝑥𝑦
Simplify each of the following:
log log 4 5
log log 3 11
log log 5 𝑎
Solve for 𝑥:
log log 3 𝑥 =2
log 2 𝑥 + log 2 𝑥−3 =2
log 𝑥 log 𝑥 16 =3

5. Solve Problems Using log 𝑎 𝑥 𝑦
Given that log 𝑎 𝑐=𝑏 and 𝑎 𝑝 𝑎 𝑞 = 𝑎 𝑝−𝑞 , prove that log 𝑎 𝑥 𝑦 = log 𝑎 𝑥 − log 𝑎 𝑦 .
Again, let log 𝑎 𝑥 =𝑝 and log 𝑎 𝑦 =𝑞

6. Solve Problems Using log 𝑎 𝑥 𝑦
Simplify each of the following:
log − log 2 25
log − log 5 5
log − log 2 4
Solve for 𝑥:
log − log 2 𝑥 =3
log 3 (𝑥+2) − log 3 𝑥 =2
log 𝑥 log 𝑥 6 =3

7. Solve Problems Using log 𝑎 𝑥 𝑞
Given that log 𝑎 𝑐=𝑏 and 𝑎 𝑝 𝑞 = a pq , prove that
log 𝑎 𝑥 𝑞 =𝑞𝑙𝑜 𝑔 𝑎 𝑥
Let log 𝑎 𝑥=𝑝

8. Solve Problems Using log 𝑎 𝑥 𝑞
Simplify each of the following:
log
log
log
Solve for 𝑥:
log 4 𝑥 2 =6
log 𝑥 =12
log 𝑥 3 𝑥 = 1 3

9. Solve Problems Using log 𝑎 𝑎 and log 𝑎 1
Using log 𝑎 𝑐=𝑏 and 𝑎 1 =𝑎, prove that log 𝑎 𝑎 =1.
Using log 𝑎 𝑐=𝑏 and 𝑎 0 =1, prove that log 𝑎 1 =0

10. Solve Problems Using log 𝑎 𝑥 = log 𝑏 𝑥 log 𝑏 𝑎
Write each of the following as a quotient of log 10 :
log 5 16
log 2 25
log
log 𝑎 𝑏
log 𝑐 10
log 5 𝑝 − log 5 𝑞

11. Solve Problems Using Logarithms
Solve for 𝑥:
log 𝑏 30 − log 𝑏 = log 𝑏 𝑥
2 log 𝑏 log 𝑏 9 − log 𝑏 3 = log 𝑏 𝑥
log 𝑏 log 𝑏 𝑥 2 = log 𝑏 𝑥
log 𝑏 (𝑥+2) − log 𝑏 4 = log 𝑏 3𝑥
Given that 𝑝= log 𝑐 𝑥 , express log 𝑐 𝑥 + log 𝑐 (𝑐𝑥) in terms of 𝑝.

12. Solve Problems Using Logarithms
log 𝑒 is frequently used in science as many natural phenomena grow or shrink according to 𝑒 – hence it is called the natural exponential.
log 𝑒 is given the special abbreviation ln .
The streptococci bacteria population 𝑁 at a time 𝑡 (in months) is given by 𝑁= 𝑁 0 𝑒 2𝑡 , where 𝑁 0 is the initial population. If the initial population was 100 in a sample, how long does it take for the population to reach 1 million?