13. Exponentials & Logarithms
High SchoolExponential functions model growth that feeds on itself; logarithms are their inverse — the tool for undoing exponents and taming enormous ranges. Together they describe populations, interest, sound and earthquakes.
Exponential growth and the logarithm as inverse
An exponential function is y = a·bˣ: the variable sits in the exponent, so the output multiplies by b for every unit increase in x. If b > 1 it grows; if 0 < b < 1 it decays. The special base e ≈ 2.71828 gives the natural exponential eˣ.
The logarithm answers the reverse question — "to what power must I raise the base?"
Because it inverts exponentiation, the logarithm turns products into sums — the log laws:
log(x/y) = log x − log y
log(xⁿ) = n·log x
Compound interest and continuous growth are captured by A = P(1 + r/n)^(nt) and, in the continuous limit, A = Pe^(rt).
Repeated multiplication, and its undo button
Linear growth adds the same amount each step; exponential growth multiplies by the same factor each step. That difference is staggering over time — the famous rice-on-a-chessboard story, where doubling each square reaches over 18 quintillion grains, is pure exponential growth. Anything that grows in proportion to its current size (money earning interest, a spreading rumour, a bacterial colony) is exponential.
Logarithms exist because we constantly need to run exponentials backwards: "the investment grew to triple — how many years did that take?" is a logarithm question. They also compress scale: the decibel, pH and Richter scales are logarithmic, so a jump of one unit means a ten-fold change. That is how a single readable number can span from a whisper to a jet engine.
Worked example: doubling time
An investment grows at 6% per year, compounded annually. How long until it doubles?
- Model: A = P(1.06)ᵗ. Doubling means A = 2P, so (1.06)ᵗ = 2.
- Take logs of both sides: t · log(1.06) = log 2.
- Solve: t = log 2 / log 1.06 ≈ 0.3010 / 0.02531 ≈ 11.9 years.
- Quick check — the "Rule of 72": 72 / 6 = 12 years, close to our exact 11.9. ✓
Answer: about 11.9 years.
Test your understanding
What is log₂ 32?
Ask: 2 to what power gives 32? Since 2⁵ = 32, the answer is log₂ 32 = 5. Logarithms simply read off the exponent.
Solve 3^(x) = 20 for x.
Take logs: x = log 20 / log 3. Numerically ≈ 1.301 / 0.477 ≈ 2.73. Check: 3^2.73 ≈ 20. ✓ Any base of log works as long as you use it on both top and bottom.
Why is log(0) undefined and log of a negative undefined (for real numbers)?
log_b y asks for the power that gives y. A positive base raised to any real power is always positive and never zero — it approaches 0 as the exponent goes to −∞ but never reaches it. So no real exponent yields 0 or a negative, leaving log 0 and log(negative) undefined over the reals.
What makes e special as a base?
e is the base for which the exponential function equals its own rate of change: the slope of eˣ at every point is eˣ itself. It arises naturally as the limit of (1 + 1/n)ⁿ as n → ∞ — the ideal of continuous compounding. This self-derivative property makes e the natural language of growth and, later, of calculus.
How the ideas connect
Every key idea in this chapter, branching from the core concept — use it to see the whole picture at a glance.
The key facts, visualised
Worked problems, step by step
Follow each solution line by line, then try to reproduce it on paper before moving on.
Example 1Solve 2^x = 32.
- Write 32 as a power of 2: 32 = 2^5
- So 2^x = 2^5
- Therefore x = 5
Example 2A colony doubles every 3 hours. How long to grow 8-fold?
- 8 = 2^3, so three doublings are needed
- Each doubling takes 3 hours
- 3 x 3 = 9 hours
Now you try
Work each one out first, then tap to reveal the worked answer.