Practice Maths

L36 — Solving Exponential Equations

Key Terms

Exponential equation
An equation where the unknown appears in the exponent, e.g. 2x = 32.
Same-base method
Write both sides as powers of the same base and equate the indices — works when both sides can be expressed as a power of the same number.
Logarithm method
Take log of both sides and use the power rule to bring the exponent down — works for any positive base.
Growth model
y = a × bx where b > 1 — the value increases exponentially over time.
Decay model
y = a × bx where 0 < b < 1 — the value decreases toward (but never reaches) zero.
Half-life
The time for a quantity to halve; found by solving bt = 0.5 for t.

Two methods for solving exponential equations

MethodWhen to useExample
Same base Both sides can be written as powers of the same base 2x = 16 ⇒ 2x = 24 ⇒ x = 4
Take logarithms Bases cannot be matched; need an exact decimal answer 3x = 20 ⇒ x = log(20)/log(3) ≈ 2.73

Taking logs of both sides

For ax = b (where a, b > 0, a ≠ 1):

x = log(b) / log(a)  =  loga(b)

Use base 10 (log) or base e (ln) — both give the same answer.

Growth and decay models

ModelFormulaKey feature
Exponential growthA = A0 × rt, r > 1Increasing without bound
Exponential decayA = A0 × rt, 0 < r < 1Approaching zero
Compound interestA = P(1 + r/n)ntGrowth per compounding period
0 2 4 6 8 16 −3 −2 −1 0 1 2 3 4 x y = 2ˣ y = (½)ˣ (0, 1) y
Growth (blue) and decay (red) — both pass through (0, 1) and never cross the x-axis.
Hot Tip: When both sides can't be written as the same base, take log of both sides and apply the power rule: log(ax) = x ⋅ log(a). This brings the unknown out of the exponent so you can solve algebraically.

Worked Example 1 — Same-base method

Solve: (a) 3x = 81   (b) 22x−1 = 32

(a) 81 = 34, so 3x = 34 ⇒ x = 4.

(b) 32 = 25, so 22x−1 = 25 ⇒ 2x − 1 = 5 ⇒ 2x = 6 ⇒ x = 3.

Worked Example 2 — Logarithm method

Solve: 5x = 200.

Take log of both sides: log(5x) = log(200).

x⋅log(5) = log(200).

x = log(200)/log(5) = 2.301/0.699 ≈ 3.29.

Worked Example 3 — Growth model

A town has 5 000 people and grows at 4% per year. When will the population reach 8 000?

5000 × 1.04t = 8000 ⇒ 1.04t = 1.6.

t⋅log(1.04) = log(1.6).

t = log(1.6)/log(1.04) = 0.2041/0.01703 ≈ 12.0 years.

Worked Example 4 — Decay model

A car depreciates at 15% per year. It cost $30 000 new. When will its value fall below $10 000?

30 000 × 0.85t < 10 000 ⇒ 0.85t < 1/3.

t⋅log(0.85) = log(1/3) ⇒ t = log(1/3)/log(0.85).

t = −0.4771/(−0.0706) ≈ 6.76 years ⇒ after 7 years.

Worked Example 5 — Compound interest

How long for $5 000 to grow to $8 000 at 6% p.a. compounded monthly?

8000 = 5000 × (1 + 0.06/12)12t = 5000 × 1.00512t.

1.00512t = 1.6 ⇒ 12t⋅log(1.005) = log(1.6).

12t = log(1.6)/log(1.005) = 0.2041/0.002166 ≈ 94.2.

t ≈ 94.2/12 ≈ 7.85 years.

See Answers ➔

  1. Same-base method. Fluency

    • (a) Solve 2x = 128.
    • (b) Solve 3x = 1/9.
    • (c) Solve 52x = 125.
    • (d) Solve 4x−1 = 8. (Hint: write both sides as powers of 2.)

  2. Logarithm method. Fluency

    • (a) Solve 2x = 10. Give answer to 2 decimal places.
    • (b) Solve 3x = 50. Give answer to 2 decimal places.
    • (c) Solve 1.5x = 10. Give answer to 2 decimal places.
    • (d) Solve 0.8x = 0.1. Give answer to 2 decimal places.

  3. Growth and decay. Fluency

    • (a) An investment of $2 000 grows at 5% per year. Write a formula for its value after t years.
    • (b) Find the value of the investment after 10 years.
    • (c) A car worth $25 000 depreciates at 12% per year. Write its value after t years.
    • (d) Find the car’s value after 5 years.

  4. Solving growth/decay for time. Fluency

    • (a) How many years until $3 000 doubles at 6% per year?
    • (b) A radioactive sample of 200 g decays at 5% per year. When does it reach 100 g?
    • (c) A colony of 500 bacteria doubles every 3 hours. When does it exceed 10 000?
    • (d) Using the Rule of 72, estimate the doubling time of an investment growing at 9% per year.

  5. Graph of exponential growth vs decay. Understanding

    The Key Ideas graph shows y = 2x (growth) and y = (1/2)x (decay).

    • (a) State the coordinates where the two graphs intersect.
    • (b) For what values of x is 2x > (1/2)x?
    • (c) Both graphs have the same y-intercept. State it and explain why using index laws.
    • (d) Describe the long-run behaviour of each function as x → +∞.

  6. Mixed exponential equations. Understanding

    • (a) Solve 2x+1 = 3x. (Take log of both sides.)
    • (b) Solve 4x = 6. Give your answer using log base 2.
    • (c) Solve 9x − 10 ⋅ 3x + 9 = 0. (Hint: let u = 3x.)
    • (d) Solve 22x = 5 × 2x − 4. (Hint: let u = 2x.)

  7. Compound interest. Understanding

    • (a) $4 000 is invested at 8% p.a. compounded quarterly. Write the formula for its value after t years.
    • (b) Find its value after 5 years.
    • (c) How many years until it reaches $10 000?
    • (d) Compare: if instead the interest is compounded annually, what is the value after 5 years?

  8. Applications of decay. Understanding

    • (a) A substance has a half-life of 10 years. Write a decay formula starting at 500 mg.
    • (b) How much remains after 25 years?
    • (c) When does it fall below 10 mg?
    • (d) Sketch the general shape of a half-life decay curve and state two key features.

  9. Newton’s Law of Cooling. Problem Solving

    A hot drink cools according to T(t) = 20 + 70 × 0.95t, where T is temperature (°C) and t is time in minutes.

    • (a) Find the initial temperature (t = 0).
    • (b) What does the “20” represent in context?
    • (c) Find the temperature after 10 minutes.
    • (d) Solve for when the drink reaches 40°C.

  10. Carbon-14 dating. Problem Solving

    Carbon-14 has a half-life of 5730 years. A fossil has 30% of its original C-14 remaining.

    • (a) Write the decay equation A(t) = A0 × (1/2)t/5730.
    • (b) Set A(t)/A0 = 0.30 and solve for t.
    • (c) Round your answer to the nearest hundred years.
    • (d) If a different fossil has 70% remaining, is it older or younger? Find its age.