Section 14, easy algebra

Indices, Surds & Logarithms

These are rule-based chapters: learn about ten rules and most questions fall in under two minutes. Rewrite everything in prime bases for indices, and convert to a single base for logs.

How CAT asks it

Solve an exponential equation by matching bases; simplify a power expression; compare the sizes of surds or powers; simplify nested roots or telescoping sums; evaluate or solve log expressions, often through a change-of-base chain; count digits using log 2 and log 3.

Concepts you must know

Indices and surds

  • aᵐ × aⁿ = aᵐ⁺ⁿ; aᵐ ÷ aⁿ = aᵐ⁻ⁿ; (aᵐ)ⁿ = aᵐⁿ; a⁰ = 1; a⁻ⁿ = 1/aⁿ; a^(1/n) is the n-th root of a. (ab)ⁿ = aⁿbⁿ.
  • If aˣ = aʸ (a > 0, a ≠ 1), then x = y. If aˣ = bˣ with a ≠ b (both positive), then x = 0.
  • Rewrite in prime bases: 8 = 2³, 27 = 3³, 125 = 5³, 0.25 = 2⁻².
  • Rationalise: 1/(√a + √b) = (√a − √b)/(a − b).
  • Nested roots: √(a + 2√b) = √x + √y when x + y = a and xy = b.
  • Compare surds by raising them to a common power: for 2^(1/2) and 3^(1/3), raise both to the 6th power and compare 8 with 9.

Logarithms

  • log_a x = y means aʸ = x, with a > 0, a ≠ 1 and x > 0.
  • log(xy) = log x + log y; log(x/y) = log x − log y; log(xⁿ) = n log x.
  • log_a b × log_b c = log_a c, and log_a b = 1/log_b a. Also a^(log_a x) = x.
  • log₁₀ 2 ≈ 0.301 and log₁₀ 3 ≈ 0.477. A number N has floor(log₁₀ N) + 1 digits.
  • Domain: the argument must be positive and the base positive and not 1. Check every solution against this.
  • Change of base, and powers in the base or argument:
log⁡ab=log⁡blog⁡a\log_a b = \frac{\log b}{\log a}
log⁡ambn=nmlog⁡ab\log_{a^m} b^{n} = \frac{n}{m}\log_a b

Thought process

  1. Indices: write every number in prime bases, then equate the exponents.
  2. Surds: rationalise, or look for a perfect-square pattern.
  3. Logs: bring everything to one base and turn sums of logs into the log of a product.
  4. Solve, then reject any answer that breaks the domain.

Patterns CAT repeats

Numbered so your error log can point at them: “Pattern 2”.

  1. 1Exponential equations solved by matching bases.
  2. 2Simplifying powers through prime factorisation.
  3. 3Comparing surds or powers.
  4. 4Telescoping surd sums and nested roots.
  5. 5Evaluating log expressions with the product, quotient and power rules.
  6. 6Log equations, with invalid roots rejected.
  7. 7Change-of-base chains.
  8. 8Counting digits with log 2 and log 3.
  9. 9Log equations that become quadratics with t = log x.

Worked examples

Attempt each one before opening its solution, then compare methods.

  1. Example 1

    Solve 2^(x + 3) = 32.

    Show solution

    32 = 2⁵, so x + 3 = 5. Answer: x = 2.

  2. Example 2

    Solve 9ˣ = 27^(x − 1).

    Show solution

    3^(2x) = 3^(3x − 3), so 2x = 3x − 3. Answer: x = 3.

  3. Example 3

    Simplify 81^(3/4) × 32^(−2/5).

    Show solution

    81^(3/4) = 3³ = 27 and 32^(−2/5) = 2⁻² = 1/4. Answer: 27/4.

  4. Example 4

    Which is largest: 2^(1/2), 3^(1/3) or 6^(1/6)?

    Show solution

    Raise all three to the 6th power: 8, 9 and 6. Answer: 3^(1/3).

  5. Example 5

    Evaluate 1/(√1 + √2) + 1/(√2 + √3) + … + 1/(√80 + √81).

    Show solution

    Rationalising, each term is √(n + 1) − √n, so the sum telescopes to √81 − √1. Answer: 8.

  6. Example 6

    Simplify √(7 + 2√10).

    Show solution

    Find two numbers that add to 7 and multiply to 10: 5 and 2. Answer: √5 + √2.

  7. Example 7

    Evaluate log₂ 8 + log₃ 81 − log₅ 125.

    Show solution

    3 + 4 − 3. Answer: 4.

  8. Example 8

    Solve log₂ x + log₂ (x − 2) = 3.

    Show solution

    x(x − 2) = 2³ = 8, so x² − 2x − 8 = 0 and x = 4 or −2. Reject −2, which makes both arguments negative. Answer: x = 4.

  9. Example 9

    Evaluate log₂ 3 × log₃ 4 × log₄ 5 × … × log₆₃ 64.

    Show solution

    The chain collapses to log₂ 64. Answer: 6.

  10. Example 10

    How many digits does 2⁵⁰ have? (log 2 = 0.301)

    Show solution

    50 × 0.301 = 15.05, so 2⁵⁰ lies between 10¹⁵ and 10¹⁶. Answer: 16.

  11. Example 11

    Solve log_x 4 + log₄ x = 5/2.

    Show solution

    Let t = log₄ x, so log_x 4 = 1/t. Then t + 1/t = 5/2, giving 2t² − 5t + 2 = 0 and t = 2 or 1/2. Answer: x = 16 or x = 2.

Traps

  • log(a + b) is not log a + log b.
  • Keeping roots that make a log’s argument zero or negative.
  • (aᵐ)ⁿ is a^(mn), but a^(mⁿ) is different: (2³)² = 2⁶ while 2^(3²) = 2⁹.

Practice set

Timed, about 2.5 minutes per question. Log every miss in the error log.

  1. 1

    Solve 5^(2x − 1) = 125.

    Show answer

    x = 2: 125 = 5³, so 2x − 1 = 3.

  2. 2

    Simplify 16^(3/4) + 27^(2/3).

    Show answer

    17: 8 + 9.

  3. 3

    Which is larger, 3^(1/3) or 4^(1/4)?

    Show answer

    3^(1/3): raise both to the 12th power and compare 81 with 64.

  4. 4

    Evaluate log₂ (1/32) + log₉ 27.

    Show answer

    −3.5: log₂ (1/32) = −5 and log₉ 27 = 3/2.

  5. 5

    Solve log₃ (x + 1) + log₃ (x − 1) = 1.

    Show answer

    x = 2: x² − 1 = 3, and x = −2 makes the arguments negative.

  6. 6

    Given log 2 = 0.301, how many digits does 8²⁰ have?

    Show answer

    19: 8²⁰ = 2⁶⁰, and 60 × 0.301 = 18.06.

Log a missFind this chapter’s real CAT questions