Lecture 3 · Barnett 1-5 to 1-6 · about 45 min

Exponential and Logarithmic Functions

Graphs of exponential and log functions with their asymptotes, compound and continuous interest, the log laws, and how to solve exponential and log equations, including for time or rate.

By the end you can

Sketch exponential and logarithmic graphs, including shifts, with the correct asymptote
Compute compound, continuous and present values
Convert between log and exponential form and apply the log laws
Solve exponential equations (same base or with ln) and log equations, rejecting answers outside the domain

Key idea · Exam recipe (memorise this)

Graph $y = b^x$ : points $(0, 1)$ , $(1, b)$ , $(-1, \frac1b)$ ; asymptote $y = 0$ . $y = b^x + k$ : asymptote $y = k$ . Graph $\ln x$ : $(1, 0)$ , $(e, 1)$ ; asymptote $x = 0$ . $\ln(x - h)$ : asymptote $x = h$ , domain $x \gt h$ .
Log ↔ exponential: $\log_b x = y \iff b^y = x$ . So $\log_2 8 = 3$ , $\log 0.01 = -2$ , $\ln 1 = 0$ , $\ln e = 1$ .
Same base: write both sides as powers of one base, set exponents equal ( $2^{x+1} = 8 = 2^3 \Rightarrow x = 2$ ).
Different bases: isolate the power, take $\ln$ of both sides, bring the exponent down, divide.
Log equation: combine into one log, convert to exponential form, solve, then check every answer in the original (arguments must be $\gt 0$ ).
Money: $A = P(1 + \frac{r}{n})^{nt}$ , continuous $A = Pe^{rt}$ , present value $P = \frac{A}{(1 + r/n)^{nt}}$ or $Ae^{-rt}$ . Solve for $t$ or $r$ with $\ln$ .

How the exam asks it: three core exam templates

In midterm exams, exponential and logarithmic questions test three canonical structures: interest compounding and investment growth horizons, solving exponential equations using matching bases or logarithms, and condensing logarithmic expressions while identifying extraneous roots. Master these three worked solutions, then follow the cheat-sheet and 4-step method below.

Exam template 1: Compound interest, continuous growth, and time to target

Question: An investor places $10{,}000$ TL in a fixed-income fund paying an annual interest rate of 6%.
a) Calculate the accumulated balance after 5 years if interest compounds annually ( $n = 1$ ).
b) Calculate the accumulated balance after 5 years if interest compounds continuously.
c) Separately, suppose 8,000 TL is deposited at 7% compounded annually. Set up the equation and solve for the time $t$ (in years) required for the fund to reach 12,000 TL.
d) Using the continuous doubling formula $t = (\ln 2)/r$ , calculate how many years it takes an economy growing at 4% continuously to double in size.

Exam template 2: Solving exponential equations (common base vs isolate and ln)

Question: Solve the following exponential equations for $x$ :
a) $3^{2x - 1} = 27$
b) $5e^{0.2x} = 40$ (round to two decimal places).

Exam template 3: Solving logarithmic equations and discarding extraneous roots

Question: Solve the equation $\log_2 x + \log_2(x - 2) = 3$ for all real solutions, and explain why any prospective solution is accepted or rejected.

Cheat-sheet & the 4-step method

· Cheat-sheet: exponential growth, logarithmic laws, and equation solving

Operation / Formula	Mathematical Rule	When to use it	How to write the exam answer
Periodic compound interest	$A = P\left(1 + \dfrac{r}{n}\right)^{nt}$	Compounding $n$ times/year ( $n=1, 4, 12$ )	State $P, r, n, t$ , compute factor, multiply by $P$ with currency units
Continuous compounding	$A = P e^{rt}$	When problem states “compounded continuously”	Evaluate $e^{rt}$ on calculator, multiply by principal $P$
Continuous doubling time	$t = \dfrac{\ln 2}{r} \approx \dfrac{0.693}{r}$	Finding time for investment to double	Divide $0.693$ by annual rate $r$ (in decimal form)
Solve exponential: same base	$b^{u} = b^{v} \implies u = v$	Both sides expressible as powers of integer $b$	Express both sides with base $b$ , equate exponents, solve linear equation
Solve exponential: base $e$	$A e^{kt} = B \implies e^{kt} = \dfrac{B}{A} \implies kt = \ln\left(\dfrac{B}{A}\right)$	Growth models with unknown in exponent	Divide by $A$ first, apply $\ln$ , divide by $k$
Log product law	$\log_b(u \cdot v) = \log_b u + \log_b v$	Condensing sum of logs with matching base	Combine into $\log_b(uv)$ before exponentiating
Log quotient law	$\log_b\left(\dfrac{u}{v}\right) = \log_b u - \log_b v$	Condensing difference of logs	Combine into $\log_b(u/v)$
Log power law	$\log_b(u^k) = k \log_b u$	Bringing exponents down to linear level	Move exponent $k$ in front as multiplier
Log to exponential conversion	$\log_b x = y \iff x = b^y$	Solving log equation after single log is isolated	Rewrite as $x = b^y$ and solve for variable

Key idea · The 4-step method for any exam question on exponentials and logarithms

Step 1: Identify the position of the variable.
- If in the exponent: prepare to isolate the power and apply logarithms (or match bases).
- If inside a logarithm: prepare to combine logs and convert to exponential form.
Step 2: Isolate the core expression.
- For exponentials: isolate $b^{f(x)}$ by dividing by any outside coefficients first (do not multiply base by outside constant!).
- For logs: use product/quotient laws to combine multiple logs into a single $\log_b(g(x))$ .
Step 3: Invert the operation.
- Exponent with base $e$ : take $\ln$ of both sides ( $\ln(e^u) = u$ ).
- General exponential $b^x = C$ : take $\ln$ of both sides ( $x \ln b = \ln C \implies x = \ln C / \ln b$ ).
- Logarithm $\log_b(g(x)) = c$ : rewrite in exponential form $g(x) = b^c$ .
Step 4: Solve, state the bold answer with units, and test for extraneous roots.
- For every log equation: substitute solutions back into original expressions to ensure all arguments are strictly positive ( $u \gt 0$ ).

Exponential functions

An exponential function multiplies by the same factor every period: compound interest, population growth, inflation.

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.

Property of $y = b^x$	Value
Domain	all reals
Range	$(0, \infty)$ : never 0, never negative
y-intercept	$(0, 1)$
Asymptote	$y = 0$ (left side if $b \gt 1$ , right side if $0 \lt b \lt 1$ )
Shape	increasing if $b \gt 1$ , decreasing if $0 \lt b \lt 1$

Compound and continuous interest

Compound interestFormula sheet →

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

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

Use it when interest compounds a fixed number of times a year: yearly n = 1, quarterly n = 4, monthly n = 12.

Continuous compoundingKey formulaFormula sheet →

A = P\,e^{rt}

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

Use it when the question says compounded continuously.

$e$ is the limit of $(1 + \frac1n)^n$ as $n \to \infty$ : compounding more and more often approaches $Pe^{rt}$ but never beats it.

Logarithms

A logarithm answers “to what power?”. It is the inverse of the exponential function.

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

Use it when you convert between log form and exponential form: the first step in almost every log question.

$\log x$ means base 10, $\ln x$ means base $e$ . Logs of 0 or negative numbers do not exist.

Log lawsRulesFormula sheet →

\log_b(MN) = \log_b M + \log_b N \qquad \log_b\!\frac{M}{N} = \log_b M - \log_b N \qquad \log_b M^p = p\log_b M

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

Use it when you expand or combine logs, or bring an exponent down to solve for it.

Special values and change of baseFormula sheet →

\log_b 1 = 0, \quad \log_b b = 1, \quad \log_b b^x = x, \quad b^{\log_b x} = x, \quad \log_b x = \frac{\ln x}{\ln b}

$b$: the base
$x$: a positive number

Use it when you simplify expressions like ln(e^{3x}) or need log_3 20 on a calculator with only ln.

Solving exponential equations

Doubling time (continuous)Formula sheet →

t_{\text{double}} = \frac{\ln 2}{r} \approx \frac{0.693}{r} \approx \frac{70}{100r}

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

Use it when you need the doubling time, or the rule of 70 for a quick estimate.

Solving log equations

Exam practice

Summary and review

$y = b^x$ : through $(0, 1)$ , asymptote $y = 0$ ; $\ln x$ : through $(1, 0)$ , asymptote $x = 0$ ; shifts move the asymptote.
$A = P(1 + r/n)^{nt}$ ; $A = Pe^{rt}$ ; present value divides instead of multiplies.
$\log_b x = y \iff b^y = x$ ; log laws only for products, quotients, powers inside the log.
Exponential equations: same base → equal exponents; otherwise isolate, $\ln$ both sides.
Log equations: combine, convert, solve, check the domain.