Logarithms and the laws of logarithms (QCE Math Methods Unit 2)

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.

Because they are inverses, the graphs of $y = b^x$ and $y = \log_b x$ are reflections of each other in the line $y = x$: the exponential's horizontal asymptote $y = 0$ becomes the logarithm's vertical asymptote $x = 0$, and the point $(0, 1)$ on the exponential becomes $(1, 0)$ on the logarithm. The logarithm is defined only for positive arguments, which is the source of the mandatory domain check on any solution.

Two bases dominate:

• Common log ($\log_{10}$, written $\log$): scientific notation, decibels, pH.
• Natural log ($\log_e$, written $\ln$): calculus and continuous growth, because $e$ is the base for which the exponential is its own derivative.

The laws of logarithms

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

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, and it is why the logarithm is the natural inverse for solving exponential growth and decay problems.

Why logarithms solve exponential equations

An exponential equation hides the unknown in the exponent, where ordinary algebra cannot reach it. Taking a logarithm of both sides uses the power law $\log(b^x) = x\log b$ to bring the exponent down to ground level as a coefficient, turning the equation into a linear one. This is the single most important use of logarithms in Methods, and it underlies every "find the time" question in growth and decay modelling.

Solving logarithmic equations

Equations containing logarithms are solved by combining the log terms into one logarithm, then converting to exponential form using the definition. For $\log_2 x + \log_2(x + 2) = 3$, combine to $\log_2[x(x + 2)] = 3$, convert to $x(x + 2) = 2^3 = 8$, solve the resulting quadratic, and finally discard any root that makes a logarithm's argument non-positive. That domain check is not optional: a value that satisfies the algebra but produces $\log$ of a negative number or zero must be rejected.

How this appears in assessment

In IA1 you may be asked to simplify a multi-term log expression with the laws or to solve an exponential equation. In the external assessment, Paper 1 tests log values and law applications, often calculator-free, while Paper 2 sets a continuous compound-interest or population-growth context where $\ln$ appears and you solve for time. Across all of these, the recurring skills are applying the three laws correctly and bringing an unknown exponent down with a logarithm.