This course starts from a simple question: to what power must a base be raised to give a number? From that definition it builds, with full proofs, the twelve laws of logarithms: products, quotients, powers, change of base and logarithms in a power. Students then use the laws to simplify long expressions, solve logarithmic equations while checking which answers are allowed, and prove relations between letters. The course ends with the graphs of logarithmic functions, comparing logarithms without a calculator, and common logarithms, where the characteristic and mantissa are used to count the digits of very large numbers.
Middle Years
Logarithms
What a logarithm is, the twelve laws of logarithms with proofs, simplifying expressions, solving logarithmic equations, graphs of log functions, and common logarithms with characteristic and mantissa.
- 9 lessons
Overview
Lessons
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What a logarithm is
The logarithm as a power: aˣ = N means x = logₐ N. Why the base must be positive and not 1, why the number must be positive, and changing between exponential and logarithmic form.
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The first laws of logarithms
Proofs and uses of logₐ 1 = 0, logₐ a = 1, the product law and the quotient law, with a warning that there is no law for the logarithm of a sum.
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Power, reciprocal and chain laws
Powers in the number and the base, swapping base and number, and the chain law that joins logarithms with matching bases, all proved from the definition.
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Change of base and logarithms in a power
The change of base law, a raised to logₐ N equals N, swapping inside a power, equal logarithms giving equal numbers, and a table of all twelve laws.
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Evaluating logarithmic expressions
A plan of attack for simplifying expressions: common bases, chains, reciprocals, roots as powers and logarithms in a power, with twelve worked examples.
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Solving logarithmic equations
Collect into one logarithm, remove it with the definition, then check every answer: the number must be positive and the base positive and not 1.
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Proving results with logarithms
Showing that a logarithmic statement follows from a given relation, and finding relations between letters from a logarithmic equation.
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Graphs of logarithmic functions
Drawing y = log₂ x and y = log x to base ½ from tables, the two shapes of y = logₐ x, and using them to find the sign of a logarithm and compare logarithms.
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Common logarithms: characteristic and mantissa
Base 10 logarithms split into characteristic and mantissa, the bar notation, counting the digits of large numbers and the zeros of small ones, and antilogarithms.
Practice papers
Ten question papers that follow the lessons in order. Each has 20 questions for 50 marks (1 hour 30 minutes), with a key sheet and worked solutions.
MCQ tests
Ten tests of 25 multiple-choice questions (30 minutes each). Answer them on the page and check your score, or download the PDF with its key and solutions.
Class notes
Logarithms
Typed from the teacher's handwritten class slides, 20 pages.
Download the notes (PDF, 818.8 KB)