Exponentials & Logarithms
Exponential growth is the most powerful force in mathematics — and logarithms are its perfect mirror image. Together they describe everything from compound interest to radioactive decay, from earthquake magnitudes to the pH of your swimming pool.
The Exponential Function: y = a^x
An exponential function multiplies by a constant factor for every unit increase in x. When the base a is greater than 1, the function grows; when a is between 0 and 1, it decays.
- a > 1: Exponential growth. The curve shoots up to the right and hugs the x-axis on the left. Every unit step to the right multiplies the value by a.
- 0 < a < 1: Exponential decay. The curve falls toward zero going right.
- a = 1: A flat horizontal line at y = 1. No growth at all.
Notice that every exponential function passes through (0, 1), because a^0 = 1 for any positive a.
Set a = 2. At x = 1, y = 2. At x = 2, y = 4. At x = 3, y = 8. The function doubles with every step. Now set a = 0.5 and watch: the function halves with every step to the right. This is exponential decay — the same pattern behind radioactive half-lives.
The Logarithmic Function: y = log_a(x)
The logarithm is the inverse of the exponential. It asks the question: “What power do I raise a to in order to get x?” In equation form:
If a^y = x, then log_a(x) = y
- The log function is only defined for positive x values.
- It passes through (1, 0) because log_a(1) = 0 for any base.
- It passes through (a, 1) because log_a(a) = 1.
- It grows slowly — painfully slowly compared to the exponential.
- It has a vertical asymptote at x = 0.
Logarithmic scales are everywhere in science. The Richter scale for earthquakes, the decibel scale for sound, and the pH scale for acidity are all logarithmic. A magnitude 7 earthquake is not “a little more” than magnitude 6 — it is 10 times more powerful.
Mirror Images: The Inverse Relationship
Exponential and logarithmic functions are inverses of each other. Graphically, this means they are reflections across the line y = x.
The yellow dashed line is y = x. The purple exponential and the red logarithm are perfect mirror images across it. If the point (2, 4) is on y = 2^x, then (4, 2) is on y = log_2(x). The coordinates simply swap.
Change the base and watch both curves adjust simultaneously. A larger base makes the exponential steeper but the logarithm flatter — they always remain reflections of each other.
Properties of Logarithms
Logarithms turn multiplication into addition, division into subtraction, and powers into multiplication. These properties are what make logs so useful:
- Product rule: log_a(MN) = log_a(M) + log_a(N)
- Quotient rule: log_a(M/N) = log_a(M) - log_a(N)
- Power rule: log_a(M^p) = p * log_a(M)
Let’s verify the product rule visually. We’ll plot log(M * x) and log(M) + log(x) with an adjustable M:
Perfect overlap. The product rule holds for every positive value of x and M.
Growth and Decay Models
In real applications, exponential functions usually appear as y = A * e^(kt) where e is approximately 2.718 (Euler’s number), t is time, and k controls the rate.
- k > 0: Growth. Population increase, compound interest, viral spread.
- k < 0: Decay. Radioactive decay, cooling, drug metabolism.
- k = 0: No change — a horizontal line at y = A.
Challenge: A bacteria colony starts with 100 cells and doubles every 3 hours. Write this as y = 100 * 2^(t/3). What is the equivalent form using base e? Hint: use the fact that 2 = e^(ln 2), so 2^(t/3) = e^(t * ln(2)/3). What is the value of k?
The Natural Logarithm
The natural logarithm ln(x) uses Euler’s number e as its base. It is the most important logarithm in calculus because it has the beautiful property that the derivative of ln(x) is 1/x.
All logarithms have the same general shape — they just differ by a constant scaling factor. In fact, log_a(x) = ln(x) / ln(a), so switching between bases is just multiplication by a constant. This is the change of base formula.
The change of base formula is why your calculator only needs one log button. To compute log_5(100), just type ln(100) / ln(5). Every logarithm can be converted to any other base.