Define logarithms as the inverse of exponentials, apply the laws of logarithms, and solve exponential equations using logarithms

What this dot point is asking

QCAA wants you to use logarithms as the inverse operation to exponentials, apply the three core log laws, and solve exponential equations that cannot be reduced to a single common base.

Definition

The logarithm $\log_b x$ is the exponent to which the base $b$ must be raised to give $x$:

Equivalently, $b^{\log_b x} = x$ and $\log_b (b^y) = y$. Logarithms and exponentials are inverse functions.

Two bases dominate:

• Common log ($\log_{10}$, written $\log$): scientific notation, decibels, pH.
• Natural log ($\log_e$, written $\ln$): calculus, continuous growth.

The laws of logarithms

For any positive base $b \neq 1$ and positive $x, y$:

DMATH_1
DMATH_2

Special values:

Change of base (useful for evaluating $\log_5 80$ on a calculator):

Solving exponential equations

If $b^x = m$ where $m$ is not a power of $b$, take log of both sides.

This is the universal method when same-base manipulation fails.

Worked example

Simplify $\log_2 24 - \log_2 3 + \log_2 4$.

Apply the quotient law to the first two, then the product law:

$\log_2 (24/3) + \log_2 4 = \log_2 8 + \log_2 4 = \log_2 (8 \times 4) = \log_2 32 = 5$ (since $2^5 = 32$).

Common traps

Treating $\log(x+y)$ as $\log x + \log y$. Not a log law. The product law applies to $\log(xy)$, not $\log(x+y)$.

Forgetting the base of $\log$. In QCAA Math Methods, $\log$ without a base usually means $\log_{10}$, and $\ln$ means $\log_e$. Be explicit when answering.

Dividing inside the log instead of subtracting outside. $\log(20/4) = \log 5$, but it can also be expanded as $\log 20 - \log 4$. The two are equal.

Taking log of a negative number. $\log_b$ is only defined for positive arguments. Equations like $\log(x-3) = 1$ require $x > 3$ in the domain.

How this appears in IA1 and EA

IA1. Simplify a multi-term log expression using the laws; solve an exponential equation.

EA Paper 1. Multiple choice on log values and law applications.

EA Paper 2. A continuous compound interest or population-growth context where $\ln$ appears, followed by solving for time.

In one sentence

The logarithm $\log_b x = y$ is defined by $b^y = x$, and the three laws ($\log_b(xy) = \log_b x + \log_b y$, $\log_b(x/y) = \log_b x - \log_b y$, $\log_b(x^n) = n \log_b x$) plus change of base let you simplify expressions and solve exponential equations whose two sides cannot be reduced to a single base.