# Logarithms and logarithmic functions · Year 12

> Logarithms and logarithmic functions, Year 12: What a logarithm is and more. Explanation, worked examples and free practice for the national curriculum in England.

In Logarithms and logarithmic functions (Year 12) you practise 5 skills: What a logarithm is · The laws of logarithms · Properties of the logarithmic function · Logarithmic equations · Transformations of a logarithmic function.

A logarithm undoes raising to a power. The log base 2 of 8 is the power you raise 2 to in order to get 8, so it is 3, because 2 to the power of 3 is 8. The base is a positive number other than 1, and the number inside the logarithm must be positive. When no base is written, the base is 10: log 1,000 is 3.

There are three laws. The log of a product is the sum of the logs, the log of a quotient is the difference of the logs, and the log of a power is the index times the log of the base. For example, log 2 plus log 5 is log 10, which is 1.

A logarithmic function is defined only for positive numbers, crosses the horizontal axis at the point (1, 0), and has the vertical axis as its asymptote. When the base is greater than 1 the function is increasing; when the base is between 0 and 1 it is decreasing.

To solve a logarithmic equation or inequality, find the domain first: everything inside a logarithm must be positive. Then write the equation as a power, or both sides of the inequality as logarithms in the same base, and keep only the solutions inside the domain. When the base is between 0 and 1, the inequality sign turns round.

**Tip:** A logarithm statement and a power statement say the same thing: the log base 2 of 8 is 3 means that 2 to the power of 3 is 8.

## What you learn

### What a logarithm is

- Example: What is the value of log₂ 16 ?
- Answer: 4
- Solution: Find the power of 2 that gives 16 : 2⁴ = 16 So log₂ 16 = 4

### The laws of logarithms

- Example: What is the value of log₂ 24 − log₂ 3 ?
- Answer: 3
- Solution: By the quotient law, the difference is the log of the quotient: log₂ (24 ÷ 3) = log₂ 8 and since 2³ = 8 , log₂ 8 = 3

### Properties of the logarithmic function

- Example: Here is the function: y = log₂ x What is its domain?
- Answer: x > 0
- Solution: What is inside the logarithm must be positive: x > 0

### Logarithmic equations

- Example: Solve the equation: log₂ (x − 4) = 4 What is the solution?
- Answer: 20
- Solution: The domain: x > 4 . Write the equation as a power: x − 4 = 2⁴ = 16 x = 20 and it is in the domain.

### Transformations of a logarithmic function

- Example: Here is the function: y = 3 log₂ x + 2 Which of these graphs is its graph? Graph A: (Graph A: a rising curve through (2, 0) and (3, 1); the asymptote is the vertical line through (1, 0).) Graph B: (Graph B: a rising curve through (1, 2) and (2, 5); the vertical axis is the asymptote.) Graph C: (Graph C: a rising curve through (1, 0) and (2, 1); the vertical axis is the asymptote.) Graph D: (Graph D: a rising curve through (1, 0) and (2, 3); the vertical axis is the asymptote.)
- Options: Graph A · Graph C · Graph D · Graph B
- Answer: Graph B
- Solution: Put two values into the formula: the graph passes through (1, 2) and (2, 5) . That is graph B.

The exercises are made by rules and the answers are worked out in code. The topic follows selected AS and A level mathematics practice in England (the GCE subject content).

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