Logarithms and Polynomials — NSW Year 10 Advanced Maths
Logarithms and their laws, graphs and equations, then polynomials: operations, division, the remainder and factor theorems, zeros and graphs. Eleven subtopics and 220 practice questions, aligned to MA5-LOG-P-01 and MA5-POL-P-01.
Logarithms and Polynomials in NSW Year 10 Advanced
This topic brings together two Stage 5 Path areas that lead straight into Year 11 Advanced. Logarithms answer the question 'what power?': \(\log_a y=x\) means \(a^x=y\). Polynomials extend quadratics to expressions such as \(x^3-6x^2+11x-6\), which can be divided, factorised using the factor theorem and graphed from their zeros.
The 11 subtopics below start with logarithm basics, graphs and laws, and solving equations with logarithms, then move to polynomials, expanding and dividing them, the remainder and factor theorems, finding zeros and sketching graphs. Logarithms and Polynomials is one of the 16 topics in NSW Year 10 Advanced Maths and sits within the Number and Algebra strand (NESA outcomes MA5‑LOG‑P‑01 and MA5‑POL‑P‑01). Both are used throughout Year 11 and 12 Advanced and Extension 1.
Subtopics
Logarithms and Polynomials: common questions
What is a logarithm?
A logarithm is the power a base must be raised to. \(\log_a y=x\) means \(a^x=y\). For example, \(\log_{10}1000=3\) because \(10^3=1000\), and \(\log_8 2=\frac{1}{3}\) because \(8^{\frac{1}{3}}=2\).
What are the log laws?
\(\log_a(xy)=\log_a x+\log_a y\), \(\log_a\left(\frac{x}{y}\right)=\log_a x-\log_a y\) and \(\log_a x^n=n\log_a x\). So \(3\log_{10}2+\log_{10}12.5=\log_{10}(8\times12.5)=\log_{10}100=2\).
How do you solve \(2^x=48\)?
Take logarithms of both sides: \(x\log2=\log48\), so \(x=\frac{\log48}{\log2}\approx5.58\).
What is the factor theorem?
If \(P(a)=0\), then \(x-a\) is a factor. For example, with \(P(x)=x^3-3x^2-4x+12\), \(P(2)=0\), so \(x-2\) is a factor and \(P(x)=(x-2)(x+2)(x-3)\).
Which NESA outcomes cover this topic?
Logarithms, including their laws, graphs and equations, are MA5‑LOG‑P‑01. Polynomials, including division, the remainder and factor theorems, zeros and graphs, are MA5‑POL‑P‑01. Both are Path outcomes in Stage 5 of the NSW Mathematics K–10 Syllabus (2022).