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CDS Mathematics · Logarithms

Logarithmic and Exponential Equations

Turn a log equation into a power (log N = k means N = 10ᵏ), or take logs of an exponential equation, and solve what remains.

All Logarithms notesCDS Mathematics strategy

Why this matters

Ten PYQs, five of them HARD — the hardest page in the chapter. Two directions: log₁₀[995 + √(…)] = 3 unwraps to 995 + √(…) = 1000; 5ˣ⁻³ = 8 needs logs of both sides. Always check that each root keeps every logarithm's argument positive.

Concept 1 of 1: Unwrapping logs and taking logs

log⁡10N=k\log_{10}N = k and N=10kN = 10^k say the same thing, so a log equation can be turned into an ordinary one. An equation with the unknown in an exponent goes the other way: take logs and the exponent comes down as a multiplier.

Definition

  • log⁡10N=k  ⟺  N=10k\log_{10}N = k \iff N = 10^k.
  • af(x)=ba^{f(x)} = b: f(x)log⁡a=log⁡bf(x)\log a = \log b.
  • Collect the logs on one side: log⁡x+log⁡x2=3log⁡x\log x + \log x^2 = 3\log x.
  • An exponential in disguise: put t=3xt = 3^x and solve the quadratic in tt.
  • Reject any root that makes a log's argument zero or negative.

Unwrapping

log⁡10N=k  ⟺  N=10k\log_{10} N = k \iff N = 10^k

Worked example

Solve log⁡10[996+x2−4x+13]=3\log_{10}\left[996 + \sqrt{x^2 - 4x + 13}\right] = 3.
Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2021 · CDS (I) 2021 — Elementary Mathematics · Q45Moderate

Example 1 · Logarithms · Solving Logarithmic Equations

If 5x−3=85^{x-3} = 8, then what is xx equal to?

Check the argument

A value that solves the algebra can make log⁡(100001−4x)\log(100001 - 4^x) the log of a negative number. Substitute each root back before choosing.
Drill 9 more on unwrapping logs and taking logs

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (1)

Watch out for (1)

Test yourself on Logarithms

15 past CDS questions from this chapter, timed at 18 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

Take the Logarithms chapter testDrill all 10 questions