Module 4: Comprehensive Study Guide

Exponential and Logarithmic Functions

Section 1: Exponential Functions

Definition

Exponential Function:

f(x) = a·b^x

where a ≠ 0, b > 0, and b ≠ 1

a = initial value (y-intercept when x = 0)
b = base (growth or decay factor)
x = exponent (independent variable)

Exponential Growth vs. Decay

Type	Condition	Behavior	Example
Growth	b > 1	Function increases as x increases	f(x) = 2(1.5)^x
Decay	0 < b < 1	Function decreases as x increases	f(x) = 2(0.5)^x

Key Properties

Domain: All real numbers, (-∞, ∞)
Range: All positive real numbers, (0, ∞)
Y-intercept: (0, a) since f(0) = a·b⁰ = a·1 = a
Horizontal Asymptote: y = 0 (the x-axis)
One-to-one: Each output corresponds to exactly one input
Continuous: No breaks or gaps in the graph

The Natural Exponential Function

f(x) = e^x

where e ≈ 2.71828... (Euler's number)

The natural exponential function is the most important exponential function in mathematics, appearing in calculus, statistics, and many applications.

Transformations

Transformation	Form	Effect
Vertical shift	f(x) = b^x + k	Shifts graph up k units (down if k < 0) New asymptote: y = k
Horizontal shift	f(x) = b^x-h	Shifts graph right h units (left if h < 0)
Vertical reflection	f(x) = -b^x	Reflects graph across x-axis
Horizontal reflection	f(x) = b^-x	Reflects graph across y-axis

Compound Interest Formulas

Periodic Compounding:

A = P(1 + r/n)^nt

A = final amount
P = principal (initial investment)
r = annual interest rate (as decimal)
n = number of times compounded per year
t = time in years

Continuous Compounding:

A = Pe^rt

Example:

Problem: Invest $1,000 at 6% annual interest compounded monthly for 3 years.

Solution:

P = 1000, r = 0.06, n = 12, t = 3

A = 1000(1 + 0.06/12)¹²⁽³⁾

A = 1000(1.005)³⁶

A ≈ $1,196.68

Section 2: Logarithmic Functions

Definition

Logarithm: The inverse of an exponential function

y = log_b(x) ⟺ b^y = x

Read as: "y equals log base b of x" means "b to the y power equals x"

where b > 0, b ≠ 1, and x > 0

Key Insight: A logarithm answers the question: "What exponent do I need on the base to get this number?"

Special Logarithms

Common Logarithm (base 10):

log(x) = log₁₀(x)

Natural Logarithm (base e):

ln(x) = log_e(x)

Basic Logarithm Properties

log_b(1) = 0 (because b⁰ = 1)
log_b(b) = 1 (because b¹ = b)
log_b(b^x) = x (inverse property)
b^log_b(x) = x (inverse property)
ln(1) = 0
ln(e) = 1

Properties of Logarithms

Product Rule:

log_b(MN) = log_b(M) + log_b(N)

The log of a product is the sum of the logs.

Quotient Rule:

log_b(M/N) = log_b(M) - log_b(N)

The log of a quotient is the difference of the logs.

Power Rule:

log_b(M^p) = p·log_b(M)

The log of a power is the exponent times the log.

Example: Expanding Logarithms

Problem: Expand ln(x²y³/z)

Solution:

ln(x²y³/z) = ln(x²y³) - ln(z) [quotient rule]

= ln(x²) + ln(y³) - ln(z) [product rule]

= 2ln(x) + 3ln(y) - ln(z) [power rule]

Example: Condensing Logarithms

Problem: Condense 3log(x) - log(y) + 2log(z)

Solution:

= log(x³) - log(y) + log(z²) [power rule]

= log(x³z²) - log(y) [product rule]

= log(x³z²/y) [quotient rule]

Change of Base Formula

log_b(x) = log_c(x) / log_c(b) = ln(x) / ln(b)

This allows you to evaluate any logarithm using a calculator (which typically has only log and ln buttons).

Example: Change of Base

Problem: Evaluate log₅(30)

Solution:

log₅(30) = ln(30)/ln(5) = 3.4012/1.6094 ≈ 2.113

Graph Characteristics

Domain: (0, ∞) - only positive numbers
Range: All real numbers, (-∞, ∞)
X-intercept: (1, 0) since log_b(1) = 0
Vertical Asymptote: x = 0 (the y-axis)
One-to-one: Each output corresponds to exactly one input
Inverse of exponential: Reflection of y = b^x across y = x

Section 3: Solving Exponential and Logarithmic Equations

Strategy 1: Exponential Equations with Same Base

If b^m = bⁿ, then m = n (one-to-one property)

Example:

Solve: 2^x+3 = 8

Rewrite 8 as 2³:

2^x+3 = 2³

Equate exponents: x + 3 = 3

Solution: x = 0

Strategy 2: Exponential Equations with Different Bases

Steps:

Isolate the exponential expression
Take the logarithm of both sides
Use the power rule to bring down the exponent
Solve for the variable

Example:

Solve: 3^x = 20

Take ln of both sides:

ln(3^x) = ln(20)

x·ln(3) = ln(20)

x = ln(20)/ln(3) = 2.9957/1.0986

Solution: x ≈ 2.727

Strategy 3: Logarithmic Equations

Steps:

Use log properties to combine into a single logarithm
Convert to exponential form or use one-to-one property
Solve the resulting equation
Check that solutions make all logarithms defined (arguments must be positive)

Example 1: Using One-to-One Property

Solve: log₂(x) + log₂(x - 3) = 2

Combine using product rule:

log₂[x(x - 3)] = 2

Convert to exponential form:

x(x - 3) = 2²

x² - 3x = 4

x² - 3x - 4 = 0

(x - 4)(x + 1) = 0

x = 4 or x = -1

Check: x = -1 makes log₂(-1) undefined, so it's extraneous.

Solution: x = 4

Example 2: Converting to Exponential Form

Solve: log₃(2x - 5) = 2

Convert to exponential form:

3² = 2x - 5

9 = 2x - 5

14 = 2x

Solution: x = 7

Common Pitfalls

Always check solutions in logarithmic equations - negative arguments are undefined
Remember: log_b(M + N) ≠ log_b(M) + log_b(N)
Remember: log_b(M - N) ≠ log_b(M) - log_b(N)
When taking logs of both sides, make sure both sides are positive

Section 4: Applications

Population Growth and Exponential Growth

P(t) = P₀e^rt

P(t) = population at time t
P₀ = initial population
r = growth rate (as decimal)
t = time

Radioactive Decay and Exponential Decay

A(t) = A₀e^-kt

A(t) = amount remaining at time t
A₀ = initial amount
k = decay constant (positive)
t = time

Half-life Formula:

t_1/2 = ln(2) / k

The time it takes for half of the substance to decay.

Newton's Law of Cooling

T(t) = T_room + (T₀ - T_room)e^-kt

T(t) = temperature at time t
T_room = ambient temperature
T₀ = initial temperature
k = cooling constant
t = time

pH Scale

pH = -log[H⁺]

where [H⁺] is the hydrogen ion concentration in moles per liter

pH < 7: acidic
pH = 7: neutral
pH > 7: basic (alkaline)

To find [H⁺] from pH:

[H⁺] = 10^-pH

Logarithmic Scales

Many real-world measurements use logarithmic scales:

Richter Scale: Earthquake magnitude
Decibel Scale: Sound intensity
pH Scale: Acidity/alkalinity

On logarithmic scales, each unit represents a multiplicative (not additive) change.

Section 5: Practice Problems with Solutions

Problem 1: Evaluate without a calculator: log₄(64)

Solution:
Ask: "4 to what power equals 64?"
4³ = 64
Answer: 3

Problem 2: Expand: ln(x²/√y)

Solution:
ln(x²/√y) = ln(x²/y^1/2)
= ln(x²) - ln(y^1/2) [quotient rule]
= 2ln(x) - (1/2)ln(y) [power rule]
Answer: 2ln(x) - (1/2)ln(y)

Problem 3: Condense: 4log(x) - 2log(y)

Solution:
= log(x⁴) - log(y²) [power rule]
= log(x⁴/y²) [quotient rule]
Answer: log(x⁴/y²)

Problem 4: Solve: 5^x = 125

Solution:
Rewrite 125 as 5³:
5^x = 5³
x = 3
Answer: x = 3

Problem 5: Solve: 2^x = 15

Solution:
Take ln of both sides:
ln(2^x) = ln(15)
x·ln(2) = ln(15)
x = ln(15)/ln(2)
x = 2.7081/0.6931
Answer: x ≈ 3.907

Problem 6: Solve: log₃(x) = -2

Solution:
Convert to exponential form:
3^-2 = x
x = 1/9
Answer: x = 1/9

Problem 7: Solve: ln(x + 2) = 3

Solution:
Convert to exponential form (base e):
e³ = x + 2
x = e³ - 2
x ≈ 20.09 - 2
Answer: x ≈ 18.09

Problem 8: A bacteria culture starts with 200 bacteria and doubles every 3 hours. Write an exponential function for the population.

Solution:
Initial population: P₀ = 200
Growth factor: doubles, so b = 2
Time unit: every 3 hours, so use t/3 as exponent
Answer: P(t) = 200·2^t/3
where t is in hours

Problem 9: How much should you invest now at 5% compounded continuously to have $10,000 in 8 years?

Solution:
Use A = Pe^rt:
10,000 = P·e^0.05(8)
10,000 = P·e^0.4
10,000 = P·1.4918
P = 10,000/1.4918
Answer: P ≈ $6,703.20

Problem 10: The pH of a solution is 5.2. Find the hydrogen ion concentration.

Solution:
pH = -log[H⁺]
5.2 = -log[H⁺]
-5.2 = log[H⁺]
[H⁺] = 10^-5.2
Answer: [H⁺] ≈ 6.31 × 10^-6 moles per liter

Summary of Key Formulas

Exponential Functions:

f(x) = a·b^x
f(x) = e^x
A = P(1 + r/n)^nt (compound interest)
A = Pe^rt (continuous compounding)

Logarithmic Functions:

y = log_b(x) ⟺ b^y = x
Product: log_b(MN) = log_b(M) + log_b(N)
Quotient: log_b(M/N) = log_b(M) - log_b(N)
Power: log_b(M^p) = p·log_b(M)
Change of base: log_b(x) = ln(x)/ln(b)

Applications:

P(t) = P₀e^rt (population growth)
A(t) = A₀e^-kt (radioactive decay)
t_1/2 = ln(2)/k (half-life)
pH = -log[H⁺]

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