Logarithmic functions: properties and equation-solving strategies

Logarithmic functions appear throughout senior secondary mathematics in Australia, from the Mathematical Methods courses in Year 11 and Year 12 to the statistics strands that feed into ATAR calculations. Students sitting the HSC in Sydney, the VCE in Melbourne, or the QCE in Brisbane all meet logarithms as a natural extension of exponential work. Understanding what a logarithm actually means, what rules govern its behaviour, and how to manipulate equations involving it is essential for confident exam performance.

The good news is that logarithmic functions follow a small set of consistent patterns. Once the foundational definition is clear, the graphs become predictable, the laws become memorisable, and even the trickiest equations reduce to a sequence of tidy algebraic steps. The sections below build that understanding from the ground up, with worked examples and Australian context woven through.

Foundations of logarithmic functions

A logarithm answers a simple question: to what power must a given base be raised to produce a given number? The statement log_b(x) = y means exactly the same thing as b^y = x. When the base is understood to be 10, the notation is usually written log x; when the base is the mathematical constant e ≈ 2.71828, it becomes ln x, the natural logarithm. Both forms appear in classroom resources used in schools from Perth to Adelaide.

The link between exponential and logarithmic functions is not just algebraic, it is visual. The graph of y = log_b(x) is the mirror image of y = b^x reflected across the line y = x. This symmetry is why logarithmic scales are used wherever quantities vary across many orders of magnitude, from the intensity of earthquakes on the Richter scale to the loudness of a crowd at an AFL match at the MCG on a packed Saturday afternoon.

For students tackling Year 11 Mathematical Methods, the priority is becoming fluent with the two bases used most often: base 10 and base e. Calculators sold in Australian retailers handle both, and the standard expectation in most state curricula is that learners can convert freely between log and ln using the change of base formula.

Core properties and logarithm laws

Three laws carry nearly all the manipulation needed in a typical school setting. The product law states that log_b(mn) = log_b(m) + log_b(n). The quotient law gives log_b(m/n) = log_b(m) − log_b(n). The power law, often the most heavily examined, says log_b(m^k) = k·log_b(m). Together these turn tangled expressions into manageable ones.

Consider an expression such as log_2(8x^3). Applying the product and power laws produces log_2(8) + 3·log_b(x), which simplifies to 3 + 3·log_2(x). The same approach works for any base, and many ATAR preparation textbooks from Brisbane-based publishers include chapters of drills built around these three identities.

The change of base formula, log_b(x) = log_c(x) / log_c(b), is the bridge between different bases. It allows a logarithm of any base to be evaluated on a calculator set to base 10 or base e, which is particularly useful when tackling VCE exam questions that permit only specific calculator modes.

Graphs, domain, range and key transformations

Every logarithm graph has a vertical asymptote at x = 0, never crosses the x-axis at a negative x-value, and rises slowly to the right when the base exceeds 1. The domain is always restricted to x > 0, while the range covers all real numbers. These features hold regardless of the base, although the steepness of the curve depends on whether the base sits above or below e.

Transformations behave in the same way as for any other graph. A vertical shift moves the asymptote, a horizontal stretch or compression changes the steepness, and a reflection in the x-axis swaps an increasing curve for a decreasing one. Students preparing for the HSC often sketch y = −2·log_b(x − 1) + 3 as practice, identifying the asymptote at x = 1, the reflection, and the vertical stretch factor.

A neat classroom check is the inverse relationship with exponentials. Spotting that y = ln(x) undoes y = e^x allows quick sketching, and the same applies to log_2(x) versus 2^x. The point (1, 0) always lies on the curve because every logarithm evaluates to zero when its argument equals 1, a useful anchor when drawing graphs by hand in a SAC or final exam.

Solving logarithmic equations step by step

Solving equations containing logarithms typically follows a three-stage process: rewrite in exponential form, isolate the variable, and reject any extraneous solution that violates the domain. The rewrite step uses the definition: if log_b(x) = y then x = b^y.

A simple case is log_3(x + 2) = 4. Converting gives x + 2 = 3^4 = 81, so x = 79. The domain check passes because 79 > −2, meaning the original logarithm is defined. For a slightly trickier problem such as log_2(x) + log_2(x − 6) = 4, the laws combine the left side into log_2(x(x − 6)) = 4, leading to x(x − 6) = 16, a quadratic that factors as (x − 8)(x + 2) = 0. Only x = 8 satisfies the domain requirement that both x and x − 6 be positive.

When extraneous solutions appear, they do so because squaring or multiplying across an equation can introduce values that make the logarithm of a negative number. Always checking each root against the original equation is non-negotiable, and examiners across all mainland states reward the discipline of writing a brief "since x > 0, accept x = 8" line of working.

Practical applications and exam preparation

Logarithms turn up wherever a quantity grows or decays exponentially. Compound interest, bacterial population growth, the cooling of a cup of tea at a Hobart café in winter, and the decay of carbon-14 in archaeology all use logarithmic models. The pH scale in chemistry is defined as −log[H⁺], so handling pH problems in the Queensland Chemistry syllabus relies entirely on logarithm work.

Acoustics and seismology offer further exam-friendly examples. The decibel scale uses log_10(p²/p²₀), and the Richter scale uses log_10(E/E₀). Both are favourite VCE and HSC short-response items because they test whether a student can translate a written definition into a logarithmic expression and then solve for an unknown.

For students aiming at the ATAR, regular timed practice with past papers from your home state is the highest-value activity available. Combine that with a short list of laws kept on a single reference card, and the unfamiliar-looking equation at the top of an exam page becomes a familiar routine within minutes.

Strategies to strengthen your logarithm work

• Memorise the product, quotient, and power laws with a worked example for each, then redraw them weekly until automatic.
• Keep a small reference card showing the change of base formula alongside two sample conversions.
• Practise four or five domain-check problems per session so that spotting extraneous roots becomes a reflex.
• Sketch the parent graph y = ln(x) or y = log(x) once a week, labelling the asymptote and the (1, 0) point.
• Work through past papers from your specific state syllabus, whether HSC, VCE, QCE, SACE, WACE or the ACT package, to familiarise yourself with the local phrasing of questions.

The most useful next step is to open a past exam paper from your state, find every logarithm question, and solve each one while writing the working exactly as you would in an exam, including the domain check on the final line.