Algebra 1.3

Algebra · 1.3  |  Exponential & Logarithmic Functions

Chapter 1 · Algebra

1.3 Exponential &
Logarithmic Functions

§ 7 Topics · Exponents · Exponential · Logarithm · Log Rules · Solving · Natural Log · e

Contents

1. Properties of Exponents
2. Exponential Functions
3. Definition of Logarithm
4. Logarithm Rules
5. Solving Exponential Equations
6. Solving Logarithmic Equations
7. Natural Logarithm $\ln x$ and $e$

01

Properties of Exponents

These rules are the foundation of everything in this chapter:

Rule	Formula
Product	$a^m \cdot a^n = a^{m+n}$
Quotient	$\dfrac{a^m}{a^n} = a^{m-n}$
Power	$(a^m)^n = a^{mn}$
Negative	$a^{-n} = \dfrac{1}{a^n}$
Fractional	$a^{1/n} = \sqrt[n]{a}$

02

Exponential Functions

An exponential function has the form:

$$f(x) = a^x \quad (a > 0,\ a \neq 1)$$

The key feature is that the variable is in the exponent, not the base.

Why $a > 0$ and $a \neq 1$?

• $a > 0$ — a negative base causes problems, e.g. $(-2)^{1/2}$ is undefined in real numbers
• $a \neq 1$ — $1^x = 1$ for all $x$, which is just a constant, not useful

Shape of the Graph

Condition	Behavior	Example
$a > 1$	Increasing — grows rapidly	$2^x$
$0 < a < 1$	Decreasing — decays toward zero	$(1/2)^x$

📌

Both graphs: pass through $(0, 1)$ since $a^0 = 1$ · have the x-axis as a horizontal asymptote · are always above the x-axis since $a^x > 0$ always.

03

Definition of Logarithm

The logarithm is simply the inverse of the exponential function:

$$\log_a b = c \iff a^c = b$$

Read $\log_a b$ as: "What power do I raise $a$ to, in order to get $b$?" (Pronounced as 'log base a of b')

Logarithm Form	Exponential Form	Answer
$\log_2 8 = ?$	$2^? = 8$	$3$
$\log_3 9 = ?$	$3^? = 9$	$2$
$\log_5 1 = ?$	$5^? = 1$	$0$
$\log_2 \frac{1}{4} = ?$	$2^? = \frac{1}{4}$	$-2$

Domain and Range of $\log_a x$

	$a^x$	$\log_a x$
Domain	All real numbers	$x > 0$ only
Range	$y > 0$ only	All real numbers

⚠️

You cannot take the log of zero or a negative number in real numbers.

04

Logarithm Rules

Let $a > 0$, $a \neq 1$, and $M, N > 0$:

Rule	Formula	Intuition
Product	$\log_a(MN) = \log_a M + \log_a N$	Exponents add when multiplying
Quotient	$\log_a\dfrac{M}{N} = \log_a M - \log_a N$	Exponents subtract when dividing
Power	$\log_a M^k = k\log_a M$	Exponent moves to front
Base identity	$\log_a a = 1$	Because $a^1 = a$
Zero identity	$\log_a 1 = 0$	Because $a^0 = 1$
Inverse	$a^{\log_a M} = M$	Log and exponential cancel out

Change of Base Formula

Calculators only compute $\log_{10}$ and $\ln$. To compute any other base:

$$\log_a b = \frac{\log_c b}{\log_c a} = \frac{\ln b}{\ln a} = \frac{\log b}{\log a}$$

Example: $\log_2 7 = \dfrac{\ln 7}{\ln 2} \approx \dfrac{1.946}{0.693} \approx 2.807$

05

Solving Exponential Equations

Strategy: Get the same base on both sides, then equate the exponents.

Case 1 — Same base directly

$$2^{x+1} = 2^5 \implies x + 1 = 5 \implies x = 4$$

Case 2 — Convert to same base

$$4^x = 8 \implies (2^2)^x = 2^3 \implies 2^{2x} = 2^3 \implies 2x = 3 \implies x = \frac{3}{2}$$

Case 3 — Cannot get same base, use logarithm

$$3^x = 7 \implies \log_3 7 = x \implies x = \frac{\ln 7}{\ln 3} \approx 1.771$$

06

Solving Logarithmic Equations

Strategy: Isolate the logarithm, then convert to exponential form.

Example 1 — Single logarithm

$$\log_2(x + 1) = 4 \implies x + 1 = 2^4 = 16 \implies x = 15$$

Example 2 — Multiple logarithms, use rules to combine

$$\log_3 x + \log_3(x-2) = 1$$ $$\log_3[x(x-2)] = 1$$ $$x(x-2) = 3^1$$ $$x^2 - 2x - 3 = 0$$ $$(x-3)(x+1) = 0$$ $$x = 3 \quad \text{or} \quad x = -1$$

⚠️

Always check for extraneous solutions — plug answers back in and reject any that make the argument of a logarithm $\leq 0$. Here $x = -1$ is rejected because $\log_3(-1)$ is undefined.

07

Natural Logarithm $\ln x$ and $e$

The Number $e$

$e$ is a special mathematical constant:

$$e \approx 2.71828\ldots$$

It is irrational — just like $\pi$, it goes on forever without repeating. It arises naturally in growth and decay problems.

Natural Logarithm

The natural logarithm is simply the logarithm with base $e$:

$$\ln x = \log_e x$$

All the same logarithm rules apply. The most used identities are:

$$\ln e = 1 \qquad \ln 1 = 0 \qquad e^{\ln x} = x \qquad \ln(e^x) = x$$

Why is $e$ Special?

The slope of $e^x$ at any point equals its own value.

This makes $e^x$ the most natural base for exponential functions in advanced mathematics, and you will fully appreciate this property in calculus.

✓

Summary Checklist

• ✓ Identify and sketch the graph of $a^x$ for $a > 1$ and $0 < a < 1$
• ✓ Convert freely between logarithmic and exponential form
• ✓ Apply all 6 logarithm rules correctly
• ✓ Use the change of base formula
• ✓ Solve exponential equations — same base method and log method
• ✓ Solve logarithmic equations and check for extraneous solutions
• ✓ Know the key identities involving $e$ and $\ln$