Lecture 3 · Barnett ch. 3 · about 45 min

Exponential and Logarithmic Functions

Compound growth, the number e, continuous compounding, and logarithms as the tool that solves for time or rate when it sits in an exponent.

By the end you can

Distinguish simple, compound and continuous growth, and compute future values for each
Explain what the number e represents and use continuous compounding
Apply the log laws to simplify exponential expressions
Solve for an unknown exponent (time or rate) using logarithms

Exponential growth

A linear function adds the same amount each period. An exponential function multiplies by the same factor each period: growth (or decay) proportional to the current level. This is the shape of compound interest, population growth, and inflation compounding.

Exponential functionFormula sheet →

f(x) = a\,b^x

$a$: initial value, f(0)
$b$: growth factor per period, b > 0, b \neq 1

Use it when a quantity grows or shrinks by the same percentage every period. b > 1 is growth; 0 < b < 1 is decay.

Compound interestFormula sheet →

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

$P$: principal
$r$: annual interest rate
$n$: compounding periods per year
$t$: years

Use it when interest compounds a fixed number of times per year, such as monthly (n=12) or quarterly (n=4).

The number e and continuous compounding

As the number of compounding periods $n$ grows without bound, $(1 + r/n)^{nt}$ approaches a limit built from a special constant, $e \approx 2.71828$ , the base of continuous growth.

Continuous compoundingKey formulaFormula sheet →

A = P\,e^{rt}

$P$: principal
$r$: annual (continuous) interest rate
$t$: years
$e$: Euler's number, \approx 2.71828

Use it when a question says continuously compounded, or you need the theoretical upper limit of compounding more and more often.

Logarithms: undoing an exponent

A logarithm answers the question “to what power?” It is the inverse of an exponential function, exactly the same relationship as a function and its inverse from the previous lesson.

Definition of a logarithmFormula sheet →

\log_b(x) = y \iff b^y = x

$b$: base, b > 0, b \neq 1
$x$: the argument, x > 0
$y$: the exponent that produces x

Use it when you need to convert between exponential form and logarithmic form, or as a first step before applying the log laws.

Two bases dominate economics: base 10 ( $\log$ ) and base $e$ (the natural logarithm, $\ln$ ).

Log lawsRulesFormula sheet →

\ln(MN) = \ln M + \ln N \qquad \ln\!\left(\frac{M}{N}\right) = \ln M - \ln N \qquad \ln(M^p) = p\,\ln M

$M, N$: positive numbers
$p$: any real exponent

Use it when a variable sits inside an exponent, or a product/quotient/power needs to be broken apart before solving.

Solving for time: when the unknown is an exponent

The whole reason to reach for logs in business math is to solve for a variable that sits in the exponent, such as the number of years needed to double an investment.

Rule of 70 (approximation)Formula sheet →

t_{\text{double}} \approx \frac{70}{100r}

$r$: continuous annual growth rate (as a decimal)
$t_double$: approximate doubling time in years

Use it when you need a quick mental estimate of doubling time without a calculator, such as for inflation or GDP growth.

Exam practice

Summary and review

Exponential function: $f(x) = ab^x$ ; $b \gt 1$ growth, $0 \lt b \lt 1$ decay.
Compound interest: $A = P(1 + r/n)^{nt}$ ; continuous compounding: $A = Pe^{rt}$ .
$e \approx 2.71828$ is the limit of compounding infinitely often.
$\log_b(x) = y \iff b^y = x$ ; $\ln$ is base $e$ .
Log laws turn products into sums, quotients into differences, and powers into coefficients.
To solve for an exponent, take $\ln$ of both sides, then use the power rule to bring the exponent down.