Logarithms & exponentials
log_b(x) answers "what power of b gives x", so the log and exponential functions are inverses: b^(log_b x) = x. The three laws — log(xy) = log x + log y, log(x/y) = log x − log y, log(xᵏ) = k log x — turn multiplication into addition, which is what makes logs the standard tool for solving equations with the unknown in the exponent.
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Method: how to approach it
The order below is what examiners expect to see, and each step carries its own marks.
- Convert between formsb^y = x is the same statement as log_b x = y. Rewriting is often the whole solution.
- Apply the laws to combine or splitCombine several logs into one before exponentiating; split a log of a product before differentiating.
- Take logs to free an exponentFor b^x = c, take logs of both sides: x log b = log c, so x = log c / log b.
- Check the domainlog is defined only for positive arguments, so every solution must make each logged expression positive.
Worked example
Solve log₂(x) + log₂(x − 2) = 3.
- Combine: log₂(x(x−2)) = 3.
- Exponentiate: x(x − 2) = 2³ = 8, so x² − 2x − 8 = 0.
- Factor: (x − 4)(x + 2) = 0, giving x = 4 or x = −2.
- Domain check: x must exceed 2 for both logs, so x = −2 is rejected.
Answer. x = 4 — the second root is extraneous because it falls outside the domain of the original logs.
Where marks get dropped
These are the specific errors that cost credit on logarithms & exponentials questions — QED's rubric penalises each of them separately.
- Writing log(x + y) = log x + log y. The law applies to products, not sums, and this is the most damaging log error.
- Skipping the domain check. Combining logs enlarges the domain, so extraneous roots appear routinely.
- Confusing ln (base e) with log (base 10 or base 2, depending on the field). State your base if there is any doubt.
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Logarithms & exponentials — frequently asked questions
What is the change of base formula?
log_b(x) = log_c(x)/log_c(b) for any valid base c. It lets you compute any log with the ln or log₁₀ on a calculator.
Why do extraneous solutions appear?
Because combining logs into one produces an equation with a larger domain than the original. Every candidate must be tested against the original expression.
Where are logs used in computer science?
Everywhere in complexity: binary search is log₂ n, balanced trees have log n depth, and information is measured in bits, which are base-2 logarithms of possibilities.
The rest of Algebra
Foundational algebra to close prerequisite gaps. Each subtopic below has its own method, worked example and mark-losing traps.
- 1Manipulating & simplifying expressions
- 2Exponent & fraction rules
- 3Factoring & polynomials
- 4Solving linear & quadratic equations
- 5Inequalities & absolute value
- 6Logarithms & exponentials
- 7Summation notation Σ & telescoping
- 8Sets of numbers ℕ, ℤ, ℚ, ℝ
- 9Modular arithmetic basics
- 10Systems of two equations & substitution
- 11Rational expressions & partial fractions
- 12Arithmetic & geometric sequences and series
- 13Function notation, domain & reading a graph
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