Exponential & Logarithmic Functions

1. Introduction

Exponential and logarithmic functions describe the mathematics of multiplicative change — the kind of change where a quantity grows or shrinks by a fixed factor over equal steps of time. Money in a savings account, a population of bacteria, the cooling of a cup of coffee, the decay of a radioactive sample, the loudness of sound, the acidity of a solution, the brightness of stars — all of these are most naturally described with exponentials and their inverses, logarithms.

The reason these functions feel different from lines is that lines change by a constant amount (add the same number each step) while exponentials change by a constant ratio (multiply by the same number each step). That difference is enormous: multiply by $2$ repeatedly and you get $1, 2, 4, 8, 16, 32, \dots$, an explosion that quickly outruns any straight line. Understanding this "constant percentage" behavior is the heart of the topic.

A logarithm is simply the question "what exponent do I need?" turned into a function. Because exponentials and logarithms are inverses, they undo each other, and that fact is the master key for solving equations in which the unknown sits up in an exponent. In this article we build both functions, derive every logarithm rule from the laws of exponents, learn the change-of-base formula, study growth and decay models, and develop a reliable toolkit for solving exponential and logarithmic equations — including how to spot and reject extraneous solutions.

You have already seen exponentials in sequences ($2^n$) and in scientific notation ($3.2 \times 10^6$). Logarithms complete the picture by letting us solve for the exponent when the base and result are known — the exact operation needed for half-life problems, earthquake magnitudes, and any question of the form "how long until…?"

2. Core Concepts

2.1 The exponential function

An exponential function has the form

$f(x) = a\,b^{x}, \qquad b > 0,\ b \neq 1,$

where $b$ is the base and $a = f(0)$ is the initial value. We require $b > 0$ so that $b^x$ is defined for all real $x$, and $b \neq 1$ because $1^x = 1$ is just a constant.

The base controls the shape:

• If $b > 1$, the function grows: as $x$ increases, $b^x$ increases, faster and faster.
• If $0 < b < 1$, the function decays: as $x$ increases, $b^x$ shrinks toward zero.

The defining feature is a constant multiplicative rate of change: increasing $x$ by $1$ always multiplies the output by $b$, because $b^{x+1} = b \cdot b^{x}$. Every exponential graph passes through $(0, a)$ and has the $x$-axis as a horizontal asymptote (it approaches but never reaches $y = 0$ for $a \neq 0$).

2.2 Exponent laws (the foundation)

Every logarithm rule is a mirror of an exponent law. Memorize these first:

$b^{m} \cdot b^{n} = b^{m+n}, \qquad \frac{b^{m}}{b^{n}} = b^{m-n}, \qquad (b^{m})^{n} = b^{mn},$ $b^{0} = 1, \qquad b^{-n} = \frac{1}{b^{n}}, \qquad b^{1/n} = \sqrt[n]{b}.$

These laws are what make it possible to rewrite $8 = 2^3$, $81 = 3^4$, and $16 = 2^4$ — the key step in "matching bases."

2.3 The natural base $e$

One base is so convenient it gets its own letter: $e \approx 2.71828$, the natural base. It arises from continuous growth: if a quantity grows continuously at rate $r$, its size after time $t$ is $P_0 e^{rt}$. The function $e^x$ has a uniquely clean rate of change (its slope at any point equals its height), which is why calculus prefers it. For now, treat $e$ as just a particular base between $2$ and $3$.

2.4 Logarithms as inverse questions

A logarithm answers: "to what power must I raise the base to get this number?" Formally,

$\log_b y = x \iff b^{x} = y.$

So $\log_2 8 = 3$ because $2^3 = 8$, and $\log_{10} 1000 = 3$ because $10^3 = 1000$. The logarithm with base $10$ is the common log, often written $\log$; the logarithm with base $e$ is the natural log, written $\ln$.

Because $\log_b$ is the inverse of $b^x$, the two compose to the identity:

$b^{\log_b x} = x \quad (x > 0), \qquad \log_b\!\left(b^{x}\right) = x \ \ (\text{all } x).$

This inverse relationship is exactly what lets us "bring an exponent down" when solving equations.

2.5 Domain, range, and asymptotes

For $f(x) = b^x$ the domain is all real numbers and the range is $y > 0$ — an exponential output is always positive. Inverting swaps these: for $g(x) = \log_b x$ the domain is $x > 0$ and the range is all real numbers. This is why the argument of a logarithm must be strictly positive: you can never take the log of zero or a negative number (within the real numbers), because no real power of a positive base produces them. The graph of $\log_b x$ has a vertical asymptote at $x = 0$.

2.6 Deriving the logarithm rules from exponent laws

The log rules are not new magic — they are the exponent laws viewed through the inverse. Let $M = b^{m}$ and $N = b^{n}$, so by definition $m = \log_b M$ and $n = \log_b N$.

• Product rule. Since $MN = b^{m}b^{n} = b^{m+n}$, taking $\log_b$ gives $\log_b(MN) = m + n = \log_b M + \log_b N$.
• Quotient rule. Since $\dfrac{M}{N} = b^{m-n}$, we get $\log_b\!\left(\dfrac{M}{N}\right) = m - n = \log_b M - \log_b N$.
• Power rule. Since $M^{k} = (b^{m})^{k} = b^{mk}$, we get $\log_b(M^{k}) = mk = k\log_b M$.

These three identities — turning products into sums, quotients into differences, and powers into coefficients — are the workhorses of the whole topic.

2.7 Change of base

Calculators usually have only $\log$ (base $10$) and $\ln$ (base $e$). To compute any other base, use the change-of-base formula:

$\log_b x = \frac{\log_c x}{\log_c b}$

for any valid new base $c$. Derivation: let $y = \log_b x$, so $b^{y} = x$. Take $\log_c$ of both sides: $y\log_c b = \log_c x$, hence $y = \dfrac{\log_c x}{\log_c b}$. In practice we pick $c = 10$ or $c = e$, e.g. $\log_2 7 = \dfrac{\ln 7}{\ln 2} \approx \dfrac{1.9459}{0.6931} \approx 2.807$.

2.8 Growth and decay models

Real-world exponential change follows two standard templates:

• Discrete (periodic) growth: $A(t) = A_0(1 + r)^{t}$, where $A_0$ is the initial amount and $r$ is the growth rate per period. A $12\%$ annual increase means $r = 0.12$, so each year multiplies by $1.12$.
• Continuous growth: $A(t) = A_0 e^{kt}$, where $k$ is the continuous rate.
• Decay: replace $r$ with a negative value or use $0 < b < 1$. A $12\%$ annual decay means multiply by $0.88$ each year: $A(t) = A_0(0.88)^{t}$.

The half-life $t_{1/2}$ of a decaying quantity satisfies $A_0 \cdot b^{t_{1/2}} = \dfrac{A_0}{2}$, so $b^{t_{1/2}} = \dfrac{1}{2}$.

2.9 The graph of $y = \log_b x$

The logarithm graph is the reflection of $y = b^x$ across the line $y = x$. Key features: it passes through $(1, 0)$ because $\log_b 1 = 0$, and through $(b, 1)$ because $\log_b b = 1$. It rises slowly for $b > 1$ and falls slowly for $0 < b < 1$. Unlike exponentials, log graphs have no horizontal asymptote but do have a vertical asymptote at $x = 0$.

2.10 Extraneous solutions in log equations

Solving a log equation often requires condensing, which implicitly assumes every argument is positive. Algebra afterward can produce values that violate this assumption. Always substitute back into the original equation and reject any solution that makes an argument $\le 0$.

Even when no squaring is involved, the algebra of combining logarithms can introduce candidate values outside the domain. Treat domain checking as a mandatory final step, not an optional afterthought.

2.11 Compound interest connection

If you deposit $P$ at annual rate $r$ compounded $n$ times per year, the balance after $t$ years is

$A = P\left(1 + \frac{r}{n}\right)^{nt}.$

As $n \to \infty$ (continuous compounding), this becomes $A = Pe^{rt}$. The limit derivation uses the famous fact that $\left(1 + \dfrac{1}{n}\right)^{n} \to e$.

Example: $5000$ at $4\%$ compounded monthly for $10$ years gives $A = 5000\left(1 + \dfrac{0.04}{12}\right)^{120} \approx 5000(1.4908) \approx 7454$.

2.12 pH and the decibel scale

Logarithmic scales compress enormous ranges into manageable numbers. pH is defined as $\text{pH} = -\log[\text{H}^+]$, so each unit decrease in pH means ten times more acidity. The decibel level of sound is $\beta = 10\log\!\left(\dfrac{I}{I_0}\right)$, where $I$ is intensity and $I_0$ is a reference level. Doubling intensity adds only about $3$ dB, not double the decibels — a consequence of the log scale.