Define a logarithm as the inverse of a power, and use the product, quotient and power laws of logarithms to simplify expressions

What this dot point is asking

SEAB wants you to understand a logarithm as the inverse of raising to a power, to switch between logarithmic and index form, and to use the three laws of logarithms (product, quotient and power) to combine or break up log expressions. Logarithms are the tool for solving equations where the unknown is in the exponent, so this dot point is the gateway to that whole family of problems.

The answer

What a logarithm is

A logarithm answers the question "to what power must the base be raised?" The definition links logs and indices:

So $\log_2 8 = 3$ because $2^3 = 8$. The base $a$ must be positive and not equal to $1$, and you can only take the log of a positive number. Being able to convert between the two forms is the single most useful skill here.

Two values worth knowing

The first holds because $a^0 = 1$; the second because $a^1 = a$.

The three laws

For the same base $a$:

In words: the log of a product is a sum, the log of a quotient is a difference, and a power comes down to the front as a multiplier. These mirror the index laws exactly, because logs and indices are inverse operations.

Combining and breaking up

Read the laws in both directions. To write $2\log x + \log 3$ as one logarithm, bring the $2$ up as a power and then use the product law. To expand $\log\dfrac{x^2}{y}$, use the quotient and power laws to get $2\log x - \log y$. Choosing the right direction is what most questions test.

Examples in context

Example 1. Decibels and pH. Sound level in decibels and the pH scale in chemistry are both logarithmic, so a tenfold change in the underlying quantity adds a fixed amount to the log scale. The product and power laws explain why each step of $10$ corresponds to one unit, which is why logs model quantities that span huge ranges.

Example 2. Setting up an equation. To solve $5^x = 20$, take logs of both sides and use the power law: $x\log 5 = \log 20$, so $x = \dfrac{\log 20}{\log 5}$. The power law is exactly what frees the exponent, linking this dot point to solving exponential equations.

Try this

Q1. Evaluate $\log_3 81$. [1 mark]

• Cue. $3^4 = 81$, so $\log_3 81 = 4$.

Q2. Write $\log 5 + \log 4$ as a single logarithm and evaluate. [2 marks]

• Cue. $\log(5 \times 4) = \log 20$.

Q3. Express $3\log x - \log y$ as a single logarithm. [2 marks]

• Cue. $\log x^3 - \log y = \log\dfrac{x^3}{y}$.