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Logarithms and Exponentials Explained

Exponential and logarithmic functions are inverses of each other, and together they describe the most important growth and decay patterns in nature, finance, and science. From compound interest to radioactive half-life to the Richter scale for earthquakes, logs and exponentials are the mathematical tools that handle quantities changing at rates proportional to their own size.

Exponential Functions

An exponential function has the form y = a^x, where a is a positive constant called the base and x is the exponent (the variable). The defining feature of exponential growth is that the quantity increases by a constant multiplicative factor over equal time intervals — not a constant additive amount. A population doubling every ten years is growing exponentially with base 2.

The most important base in mathematics is e ≈ 2.718, known as Euler’s number. The function y = e^x is its own derivative, making it the natural choice for modelling continuous growth and decay. You will meet it constantly in A-level mathematics and in physics when dealing with radioactive decay and capacitor discharge.

The graph of y = a^x (for a > 1) is always positive, always increasing, passes through (0, 1), and rises steeply to the right while approaching zero from above as x becomes very negative (asymptote at y = 0). For 0 < a < 1, the graph decreases — this is exponential decay.

What Is a Logarithm?

The logarithm is the inverse operation to exponentiation. If a^x = y, then log_a(y) = x. Read: “the log base a of y equals x.” In words, the logarithm answers the question: what power must I raise the base to in order to get this number?

For example: 2³ = 8, so log₂(8) = 3. Similarly, 10² = 100, so log₁₀(100) = 2. The common logarithm (base 10), written simply as “log”, is used in chemistry (pH scale) and engineering. The natural logarithm (base e), written “ln”, is used in calculus, science, and modelling.

Converting between index and log form is a key skill:

• Index form: a^x = N
• Log form: x = log_a(N)

Solving Exponential Equations

The power of logarithms lies in bringing the unknown down from an exponent so it can be solved algebraically. The procedure is: take logs of both sides (in any consistent base, usually 10 or e), then apply the power rule.

Example: Solve 3^x = 20.

Take log of both sides: log(3^x) = log(20). Apply the power rule: x log 3 = log 20. Divide: x = log 20 / log 3 = 1.301 / 0.477 ≈ 2.73.

Example with e: Solve e^2x = 50. Take ln of both sides: 2x = ln(50). Therefore x = ln(50)/2 ≈ 3.912/2 ≈ 1.96.

Exponential Growth and Decay

Many real-world quantities follow the model N = N₀ e^kt, where N₀ is the initial value, k is the growth constant (positive for growth, negative for decay), and t is time. Radioactive decay uses k < 0: the number of undecayed atoms decreases over time at a rate proportional to how many atoms remain. The half-life T_½ is the time for half the sample to decay; it relates to k by T_½ = ln(2)/|k|.

Compound interest follows the same model: A = P(1 + r/n)^nt for interest compounded n times per year, or A = Pe^rt for continuous compounding. Taking the log of both sides allows you to find t (how long until a savings account reaches a target balance) or r (the effective annual rate).

Logarithmic Scales

When data spans many orders of magnitude, a linear scale is impractical. Logarithmic scales compress large ranges into manageable graphs. The pH scale measures hydrogen ion concentration: pH = −log₁₀[H⁺], so a solution with pH 3 has 10 times more H⁺ ions than one with pH 4. The Richter scale for earthquake magnitude is also logarithmic: a magnitude 7 earthquake releases roughly 31 times more energy than a magnitude 6 earthquake. The decibel scale for sound intensity works similarly.

Summary

An exponential function y = a^x grows (or decays) by constant multiplicative factors. The logarithm log_a(y) = x is its inverse, answering “what power gives y?” The three log laws — product, quotient, and power rules — are the tools for simplifying expressions and solving exponential equations. Real-world applications include radioactive decay, compound interest, and logarithmic measurement scales. Logarithms also connect closely with the concepts in probability and statistics, particularly in information theory and model fitting.