Chapter 6 β Exponential and Logarithmic Functions
Exponential and logarithmic functions are inverses of each other and appear everywhere: compound interest, population growth, radioactive decay, pH scales, earthquake magnitudes, sound intensity. This chapter builds your understanding from the ground up β what these functions are, how to graph and transform them, their properties, and how to solve equations involving them.
Table of Contents
1 β Exponential Functions
- 1.1 Definition of Exponential Functions
- 1.2 The Natural Base $e$
- 1.3 Evaluating Exponential Functions
- 1.4 Compound Interest
2 β Graphs of Exponential Functions
3 β Logarithmic Functions
- 3.1 Definition β The Inverse of an Exponential
- 3.2 Common and Natural Logarithms
- 3.3 Evaluating Logarithms
- 3.4 Domain of Logarithmic Functions
4 β Graphs of Logarithmic Functions
5 β Logarithmic Properties
- 5.1 Product Rule
- 5.2 Quotient Rule
- 5.3 Power Rule
- 5.4 Change of Base Formula
- 5.5 Expanding and Condensing Logarithms
6 β Exponential and Logarithmic Equations
- 6.1 Exponential Equations (Same Base)
- 6.2 Exponential Equations (Different Bases β Using Logarithms)
- 6.3 Logarithmic Equations
- 6.4 Extraneous Solutions
7 β Exponential and Logarithmic Models
- 7.1 Exponential Growth and Decay
- 7.2 Half-Life and Doubling Time
- 7.3 Newtonβs Law of Cooling
- 7.4 Logistic Growth
8 β Fitting Exponential Models to Data
Glossary
| Term | Definition |
|---|---|
| Exponential function | $f(x) = ab^x$ where $a \ne 0$, $b > 0$, $b \ne 1$ |
| Base ($b$) | The constant being raised to a variable power |
| Natural base ($e$) | $e \approx 2.71828\ldots$; the base of the natural exponential |
| Logarithm | $\log_b(x) = y$ means $b^y = x$; the exponent to which $b$ must be raised to get $x$ |
| Common logarithm | $\log(x) = \log_{10}(x)$ |
| Natural logarithm | $\ln(x) = \log_e(x)$ |
| Asymptote | A line the graph approaches but never reaches |
| Exponential growth | Occurs when $b > 1$ (or $r > 0$); quantity increases by a fixed percentage per period |
| Exponential decay | Occurs when $0 < b < 1$ (or $r < 0$); quantity decreases by a fixed percentage per period |
| Half-life | Time for a quantity to reduce by half |
| Doubling time | Time for a quantity to double |
| Logistic growth | Growth that levels off at a carrying capacity: $f(t) = \dfrac{c}{1 + ae^{-bt}}$ |
1 β Exponential Functions
1.1 Definition of Exponential Functions
Exponential Function: A function of the form
\[f(x) = ab^x\]where:
- $a$ is the initial value ($a = f(0)$, since $b^0 = 1$)
- $b$ is the base with $b > 0$ and $b \ne 1$
- The variable $x$ is in the exponent
If $b > 1$: exponential growth.
If $0 < b < 1$: exponential decay.
Why $b \ne 1$? If $b = 1$, then $f(x) = a \cdot 1^x = a$, which is just a constant function β not exponential.
Why $b > 0$? Negative bases cause problems with non-integer exponents (e.g., $(-2)^{1/2}$ is not real).
1.2 The Natural Base $e$
Eulerβs number:
\[e = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n \approx 2.71828182845\ldots\]$e$ is irrational and transcendental (not the root of any polynomial with integer coefficients).
The function $f(x) = e^x$ is the natural exponential function. It arises naturally in:
- Continuous compound interest
- Population growth
- Radioactive decay
- Calculus (it is its own derivative: $\tfrac{d}{dx}e^x = e^x$)
1.3 Evaluating Exponential Functions
Example: For $f(x) = 3 \cdot 2^x$, evaluate:
- $f(0) = 3 \cdot 2^0 = 3 \cdot 1 = 3$
- $f(3) = 3 \cdot 2^3 = 3 \cdot 8 = 24$
- $f(-2) = 3 \cdot 2^{-2} = 3 \cdot \tfrac{1}{4} = \tfrac{3}{4}$
- $f!\left(\tfrac{1}{2}\right) = 3 \cdot 2^{1/2} = 3\sqrt{2} \approx 4.24$
1.4 Compound Interest
Compound Interest Formulas:
| Compounding | Formula |
|---|---|
| $n$ times per year | $A = P\left(1 + \dfrac{r}{n}\right)^{nt}$ |
| Continuously | $A = Pe^{rt}$ |
Where:
- $P$ = principal (initial amount)
- $r$ = annual interest rate (decimal)
- $n$ = number of compounding periods per year
- $t$ = time in years
- $A$ = final amount
Example: Invest $5000 at 6% interest for 10 years.
Monthly compounding ($n = 12$):
\[A = 5000\left(1 + \frac{0.06}{12}\right)^{12 \cdot 10} = 5000(1.005)^{120} \approx 5000(1.8194) \approx \$9097\]Continuous compounding:
\[A = 5000e^{0.06 \cdot 10} = 5000e^{0.6} \approx 5000(1.8221) \approx \$9111\]The difference is small β continuous compounding is the theoretical upper limit.
Annual Percentage Yield (APY): The effective annual rate accounting for compounding:
\[\text{APY} = \left(1 + \frac{r}{n}\right)^n - 1\]This lets you compare rates with different compounding frequencies on equal footing.
2 β Graphs of Exponential Functions
2.1 Basic Shape and Properties
For $f(x) = b^x$ with $b > 1$ (growth):
| Property | Value |
|---|---|
| Domain | $(-\infty, \infty)$ |
| Range | $(0, \infty)$ |
| $y$-intercept | $(0, 1)$ |
| $x$-intercept | None |
| Horizontal asymptote | $y = 0$ (as $x \to -\infty$) |
| Behavior | Always increasing |
| One-to-one? | Yes |
For $f(x) = b^x$ with $0 < b < 1$ (decay): Same properties except always decreasing and HA is approached as $x \to +\infty$.
Key points for any $f(x) = b^x$:
$(- 1, 1/b)$, $(0, 1)$, $(1, b)$.
These three points plus the asymptote are enough to sketch the curve.
2.2 Transformations of Exponential Functions
The general transformed form is:
\[f(x) = ab^{x-h} + k\]| Parameter | Effect |
|---|---|
| $a$ | Vertical stretch/compression; if $a < 0$, reflection over $x$-axis |
| $h$ | Horizontal shift: right if $h > 0$, left if $h < 0$ |
| $k$ | Vertical shift: up if $k > 0$, down if $k < 0$ |
| $k$ | New horizontal asymptote: $y = k$ |
Example: Graph $g(x) = -2 \cdot 3^{x+1} + 4$.
- Base function: $3^x$ (growth)
- $a = -2$: vertical stretch by 2, reflect over $x$-axis (graph flipped)
- $h = -1$: shift left 1
- $k = 4$: shift up 4; HA: $y = 4$
- $y$-intercept: $g(0) = -2 \cdot 3^1 + 4 = -6 + 4 = -2$
- The graph decreases (because of the reflection) and approaches $y = 4$ from below.
3 β Logarithmic Functions
3.1 Definition β The Inverse of an Exponential
Logarithm: The logarithm base $b$ of $x$ is the exponent to which $b$ must be raised to produce $x$:
\[\log_b(x) = y \iff b^y = x\]Equivalently: $b^{\log_b(x)} = x$ and $\log_b(b^x) = x$.
The logarithm answers the question: β$b$ to what power gives $x$?β
Converting between forms:
| Exponential form | Logarithmic form | |β|β| | $2^3 = 8$ | $\log_2(8) = 3$ | | $10^{-2} = 0.01$ | $\log_{10}(0.01) = -2$ | | $5^0 = 1$ | $\log_5(1) = 0$ | | $e^1 = e$ | $\ln(e) = 1$ | | $3^x = 7$ | $x = \log_3(7)$ |
3.2 Common and Natural Logarithms
| Notation | Base | Name | Calculator key |
|βββ-|ββ|ββ|βββββ|
| $\log(x)$ | 10 | Common logarithm | LOG |
| $\ln(x)$ | $e$ | Natural logarithm | LN |
When to use which:
- Common logs ($\log$): Decibels, pH, Richter scale β anything based on powers of 10.
- Natural logs ($\ln$): Continuous growth/decay, calculus, most scientific formulas.
3.3 Evaluating Logarithms
Think: βThe base raised to what gives me the argument?β
| Expression | Think | Answer | |β|β|β| | $\log_2(32)$ | $2^? = 32 = 2^5$ | $5$ | | $\log_3!\left(\tfrac{1}{9}\right)$ | $3^? = \tfrac{1}{9} = 3^{-2}$ | $-2$ | | $\log_{10}(1000)$ | $10^? = 1000 = 10^3$ | $3$ | | $\ln(1)$ | $e^? = 1 = e^0$ | $0$ | | $\ln(e^5)$ | $e^? = e^5$ | $5$ | | $\log_4(4)$ | $4^? = 4 = 4^1$ | $1$ | | $\log_b(1)$ | $b^? = 1 = b^0$ | $0$ (for any valid base) | | $\log_b(b)$ | $b^? = b = b^1$ | $1$ (for any valid base) |
3.4 Domain of Logarithmic Functions
The argument of a logarithm must be strictly positive:
\(\log_b(x) \text{ is defined only when } x > 0\)
Example: Find the domain of $f(x) = \log_3(2x - 6)$.
Require $2x - 6 > 0 \Rightarrow x > 3$.
Domain: $(3, \infty)$.
$\log_b(0)$ is undefined ($b^y$ can never equal 0). $\log_b(\text{negative})$ is also undefined in the reals.
4 β Graphs of Logarithmic Functions
4.1 Basic Shape and Properties
For $f(x) = \log_b(x)$ with $b > 1$:
| Property | Value |
|---|---|
| Domain | $(0, \infty)$ |
| Range | $(-\infty, \infty)$ |
| $x$-intercept | $(1, 0)$ |
| $y$-intercept | None |
| Vertical asymptote | $x = 0$ (the $y$-axis) |
| Behavior | Always increasing (for $b > 1$) |
| One-to-one? | Yes |
| Key points | $(1/b, -1)$, $(1, 0)$, $(b, 1)$ |
The graph of $\log_b(x)$ is the reflection of $b^x$ over the line $y = x$ (since they are inverses).
4.2 Transformations of Logarithmic Functions
General form: $f(x) = a\log_b(x - h) + k$
| Parameter | Effect |
|---|---|
| $a$ | Vertical stretch/compression; if $a < 0$, flip over $x$-axis |
| $h$ | Horizontal shift; new vertical asymptote: $x = h$ |
| $k$ | Vertical shift |
| Domain becomes | $(h, \infty)$ |
Example: Graph $g(x) = -2\log_3(x - 1) + 5$.
- Parent: $\log_3(x)$
- Shift right 1 β VA: $x = 1$, domain: $(1, \infty)$
- Vertical stretch by 2, reflect
- Shift up 5
- $x$-intercept: $0 = -2\log_3(x - 1) + 5 \Rightarrow \log_3(x-1) = \tfrac{5}{2} \Rightarrow x = 1 + 3^{5/2} = 1 + 9\sqrt{3} \approx 16.59$
5 β Logarithmic Properties
5.1 Product Rule
The log of a product equals the sum of the logs.
Proof sketch: Let $\log_b(M) = p$ and $\log_b(N) = q$. Then $M = b^p$, $N = b^q$, so $MN = b^{p+q}$, meaning $\log_b(MN) = p + q$. β
5.2 Quotient Rule
The log of a quotient equals the difference of the logs.
5.3 Power Rule
The log of a power brings the exponent out front.
Common errors:
- $\log(M + N) \ne \log(M) + \log(N)$ β there is no sum rule!
- $\log(M - N) \ne \log(M) - \log(N)$
- $(\log M)^n \ne n \log M$ β the power rule requires the exponent on the argument, not on the log itself.
5.4 Change of Base Formula
This lets you compute any logarithm using $\ln$ or $\log$ on a calculator.
Example: Compute $\log_5(42)$.
\[\log_5(42) = \frac{\ln 42}{\ln 5} = \frac{3.7376}{1.6094} \approx 2.322\]Check: $5^{2.322} \approx 42$ β
5.5 Expanding and Condensing Logarithms
Expanding: Use the product, quotient, and power rules to break a single log into multiple terms.
Example: Expand $\log!\left(\dfrac{x^3 \sqrt{y}}{z^2}\right)$.
\[= \log(x^3) + \log(\sqrt{y}) - \log(z^2)\]\(= 3\log(x) + \tfrac{1}{2}\log(y) - 2\log(z)\)
Condensing: Reverse the process β combine multiple log terms into one.
Example: Condense $2\ln(a) - 3\ln(b) + \tfrac{1}{2}\ln(c)$.
\(= \ln(a^2) - \ln(b^3) + \ln(c^{1/2}) = \ln\!\left(\frac{a^2\sqrt{c}}{b^3}\right)\)
6 β Exponential and Logarithmic Equations
6.1 Exponential Equations (Same Base)
One-to-One Property of Exponentials: If $b^m = b^n$, then $m = n$.
Example: Solve $4^{2x-1} = 64$.
Rewrite with the same base: $4^{2x-1} = 4^3$.
$2x - 1 = 3 \Rightarrow x = 2$.
6.2 Exponential Equations (Different Bases β Using Logarithms)
When you canβt rewrite with the same base, take a logarithm of both sides.
Example: Solve $5^x = 37$.
$\ln(5^x) = \ln(37)$
$x \ln 5 = \ln 37$
$x = \dfrac{\ln 37}{\ln 5} = \dfrac{3.6109}{1.6094} \approx 2.244$
Example: Solve $3^{2x+1} = 7^{x-2}$.
Take $\ln$ of both sides:
$(2x + 1)\ln 3 = (x - 2)\ln 7$
$2x \ln 3 + \ln 3 = x \ln 7 - 2\ln 7$
$2x \ln 3 - x \ln 7 = -2\ln 7 - \ln 3$
$x(2\ln 3 - \ln 7) = -2\ln 7 - \ln 3$
\(x = \frac{-2\ln 7 - \ln 3}{2\ln 3 - \ln 7} = \frac{-2(1.9459) - 1.0986}{2(1.0986) - 1.9459} = \frac{-4.9904}{0.2513} \approx -19.86\)
6.3 Logarithmic Equations
Strategy: Isolate the logarithm, then exponentiate (convert to exponential form).
Example 1 β Single log: Solve $\log_2(3x - 1) = 5$.
Convert: $3x - 1 = 2^5 = 32$
$3x = 33 \Rightarrow x = 11$
Check: $\log_2(33 - 1) = \log_2(32) = 5$ β
Example 2 β Multiple logs: Solve $\log(x) + \log(x + 3) = 1$.
Condense: $\log[x(x+3)] = 1$
Convert: $x(x + 3) = 10^1 = 10$
$x^2 + 3x - 10 = 0$
$(x + 5)(x - 2) = 0 \Rightarrow x = -5$ or $x = 2$
Check: $x = -5$ β $\log(-5)$ is undefined β. Reject.
$x = 2$ β $\log(2) + \log(5) = \log(10) = 1$ β.
Answer: $x = 2$.
6.4 Extraneous Solutions
Always check your solutions in the original equation! Logarithmic equations frequently produce extraneous solutions because:
- The domain of $\log_b(x)$ requires $x > 0$.
- Algebraic manipulations can introduce invalid solutions.
Any solution that makes any log argument β€ 0 must be rejected.
7 β Exponential and Logarithmic Models
7.1 Exponential Growth and Decay
Continuous Growth/Decay Model:
\[A(t) = A_0 e^{rt}\]- $A_0$: initial amount
- $r$: continuous growth rate ($r > 0$ for growth, $r < 0$ for decay)
- $t$: time
Discrete Model (percentage rate per period):
\[A(t) = A_0 (1 + r)^t\]where $r$ is the rate of change per period.
Example β Population Growth: A city of 50,000 grows at 3% per year.
\[P(t) = 50000(1.03)^t\]After 10 years: $P(10) = 50000(1.03)^{10} \approx 50000(1.3439) \approx 67,196$
When will it double? $100000 = 50000(1.03)^t$
$(1.03)^t = 2 \Rightarrow t = \dfrac{\ln 2}{\ln 1.03} = \dfrac{0.6931}{0.02956} \approx 23.4$ years.
7.2 Half-Life and Doubling Time
Half-Life ($t_{1/2}$): The time for a quantity to reduce to half its value.
\[A(t) = A_0 \left(\frac{1}{2}\right)^{t/t_{1/2}} = A_0 \cdot 2^{-t/t_{1/2}}\]Doubling Time ($t_d$): The time for a quantity to double.
\(A(t) = A_0 \cdot 2^{t/t_d}\)
Example β Radioactive Decay: Carbon-14 has a half-life of 5730 years. If a sample has 200 mg now, how much remains after 10,000 years?
\(A(10000) = 200 \left(\frac{1}{2}\right)^{10000/5730} = 200 \cdot 2^{-1.7452} = 200 \cdot 0.2983 \approx 59.7 \text{ mg}\)
Rule of 70 (quick estimate): Doubling time $\approx \dfrac{70}{\text{growth rate in percent}}$.
At 5% growth: doubling time $\approx 70/5 = 14$ periods. (Exact: $\ln 2/\ln 1.05 \approx 14.2$.)
7.3 Newtonβs Law of Cooling
where:
- $T(t) =$ temperature at time $t$
- $T_s =$ surrounding (ambient) temperature
- $T_0 =$ initial temperature of the object
- $k > 0$ is the cooling constant
Example: A cup of coffee at 200Β°F is placed in a 70Β°F room. After 5 minutes itβs 180Β°F. Find the temperature after 20 minutes.
$180 = 70 + (200 - 70)e^{-5k} = 70 + 130e^{-5k}$
$110 = 130e^{-5k} \Rightarrow e^{-5k} = \tfrac{11}{13}$
$-5k = \ln!\left(\tfrac{11}{13}\right) \approx -0.1671 \Rightarrow k \approx 0.03341$
$T(20) = 70 + 130e^{-0.03341 \cdot 20} = 70 + 130e^{-0.6682} \approx 70 + 130(0.5127) \approx 136.7Β°\text{F}$
7.4 Logistic Growth
Logistic Growth Model:
\[f(t) = \frac{c}{1 + ae^{-bt}}\]- $c =$ carrying capacity (upper limit / horizontal asymptote)
- $a$ and $b$ are positive constants
- Initial value: $f(0) = \dfrac{c}{1 + a}$
Logistic growth starts exponentially but levels off as it approaches the carrying capacity β modeling populations with limited resources, spread of diseases, adoption of technology, etc.
Example β Disease Spread: In a school of 800 students, a flu spreads according to:
\[N(t) = \frac{800}{1 + 49e^{-0.2t}}\]- Carrying capacity: 800 (eventually all could be infected)
- Initial infected: $N(0) = \dfrac{800}{1 + 49} = \dfrac{800}{50} = 16$ students
- After 10 days: $N(10) = \dfrac{800}{1 + 49e^{-2}} = \dfrac{800}{1 + 49(0.1353)} = \dfrac{800}{7.63} \approx 105$ students
8 β Fitting Exponential Models to Data
8.1 Recognizing Exponential Data
A data set follows an exponential pattern if the ratio of consecutive $y$-values is approximately constant:
\[\frac{y_2}{y_1} \approx \frac{y_3}{y_2} \approx \frac{y_4}{y_3} \approx \cdots \approx b\]This common ratio $b$ is the base of the exponential model.
| Type | Test | Pattern |
|---|---|---|
| Linear | Equal differences in $y$ | $y_2 - y_1 = y_3 - y_2$ |
| Exponential | Equal ratios in $y$ | $y_2/y_1 = y_3/y_2$ |
8.2 Exponential Regression
Exponential Regression finds the best-fit curve $y = ab^x$ (or $y = ae^{bx}$) for a data set. Most calculators and software provide:
- The parameters $a$ and $b$
- A correlation coefficient $r$ or $r^2$ for goodness of fit
To linearize: take the log of both sides of $y = ab^x$:
\[\ln y = \ln a + x \ln b\]This is linear in $x$ vs. $\ln y$, so linear regression on $(x, \ln y)$ gives the best fit:
- Slope $= \ln b \Rightarrow b = e^{\text{slope}}$
- Intercept $= \ln a \Rightarrow a = e^{\text{intercept}}$
Key Takeaways
- Exponential functions $f(x) = ab^x$: growth if $b > 1$, decay if $0 < b < 1$. Domain: all reals. Range: $(0, \infty)$ for $a > 0$.
- Logarithms are the inverse of exponentials: $\log_b(x) = y \iff b^y = x$.
- Three key log rules: Product β add, Quotient β subtract, Power β multiply. There is no rule for $\log(M + N)$.
- Change of base: $\log_b(x) = \dfrac{\ln x}{\ln b}$ β essential for calculator use.
- Solving exponential equations: Take $\ln$ (or $\log$) of both sides, then use algebra.
- Solving logarithmic equations: Condense to a single log, convert to exponential form, always check for extraneous solutions.
- Models: $A = A_0 e^{rt}$ (continuous), $A = A_0(1+r)^t$ (discrete), half-life/doubling time, Newtonβs cooling, logistic growth.
- Exponential data has constant ratios; linear data has constant differences.
Practice Questions
Q1. Evaluate without a calculator: $\log_8(512)$.
Answer:
$8^? = 512$. Since $8 = 2^3$ and $512 = 2^9$, we need $(2^3)^? = 2^9 \Rightarrow 3? = 9 \Rightarrow ? = 3$.
$\log_8(512) = 3$.
Q2. Expand: $\ln!\left(\dfrac{e^2 x^3}{y^4}\right)$.
Answer:
$= \ln(e^2) + \ln(x^3) - \ln(y^4) = 2 + 3\ln x - 4\ln y$
Q3. Solve: $3 \cdot 2^{x-1} = 48$.
Answer:
$2^{x-1} = 16 = 2^4$
$x - 1 = 4 \Rightarrow x = 5$
Q4. Solve: $\ln(x - 3) + \ln(x + 1) = \ln(3x + 1)$.
Answer:
$\ln[(x-3)(x+1)] = \ln(3x+1)$
$(x-3)(x+1) = 3x+1$
$x^2 - 2x - 3 = 3x + 1$
$x^2 - 5x - 4 = 0$
$x = \dfrac{5 \pm \sqrt{25 + 16}}{2} = \dfrac{5 \pm \sqrt{41}}{2}$
$x \approx 5.70$ or $x \approx -0.70$
Check $x \approx -0.70$: $\ln(-0.70 - 3) = \ln(-3.70)$ β undefined β. Reject.
Answer: $x = \dfrac{5 + \sqrt{41}}{2} \approx 5.70$
Q5. $10,000 is invested at 4.5% compounded quarterly. How long until it doubles?
Answer:
$20000 = 10000\left(1 + \dfrac{0.045}{4}\right)^{4t} = 10000(1.01125)^{4t}$
$2 = (1.01125)^{4t}$
$\ln 2 = 4t \ln(1.01125)$
$t = \dfrac{\ln 2}{4\ln(1.01125)} = \dfrac{0.6931}{4(0.01119)} = \dfrac{0.6931}{0.04476} \approx 15.49$ years
Q6. A substance decays continuously with a half-life of 12 days. If you start with 100 grams, how much remains after 30 days?
Answer:
Find $r$: $50 = 100e^{12r} \Rightarrow e^{12r} = 0.5 \Rightarrow 12r = \ln 0.5 \Rightarrow r = \dfrac{-0.6931}{12} \approx -0.05776$
$A(30) = 100e^{-0.05776 \cdot 30} = 100e^{-1.7329} \approx 100(0.1768) \approx 17.68$ grams.
Alternative: $A(30) = 100 \cdot \left(\tfrac{1}{2}\right)^{30/12} = 100 \cdot 2^{-2.5} \approx 100(0.1768) \approx 17.68$ grams. β