Exponents and Logarithms

MAT1501 - Fundamental Mathematics · Algebra Tools

Exponents and Logarithms

This topic covers exponents and logarithms, two important concepts in mathematics. Understanding these concepts is essential for solving equations and working with functions in algebra and calculus.

Key idea: After studying this topic, you should be able to define powers with rational exponents, apply exponential rules, convert between exponential and logarithmic forms, and solve equations involving exponents and logarithms.

1. Exponents

1.1 Rational Exponents

Exponents indicate how many times a number, called the base, is multiplied by itself. When the exponent is a rational number, it can be expressed in the form of a fraction. For example, if we have a rational exponent like a^m/n, it can be rewritten as:

a^m/n = (n√a)^m or (a^1/n)^m.

1.2 Exponential Rules

There are several rules for working with exponents:

Multiplication of powers with the same base: a^x × a^y = a^x+y
Division of powers with the same base: a^x / a^y = a^x-y
Power of a power: (a^x)^y = a^xy
Power of a product: (ab)^x = a^x b^x
Power of a quotient: (a/b)^x = a^x / b^x
Zero exponent: a⁰ = 1
Negative exponent: a^-x = 1/a^x

1.3 Examples

Let’s apply these rules in some examples:

Example 1: Simplify (2³ × 2⁴).
2³ × 2⁴ = 2³⁺⁴ = 2⁷

Example 2: Simplify (x⁵ / x²).
x⁵ / x² = x^5-2 = x³

2. Logarithms

2.1 Definition

A logarithm is the inverse operation to exponentiation. If you have an equation in exponential form, you can express it in logarithmic form. For example, if a^x = b, then log_a(b) = x.

2.2 Common and Natural Logarithms

Common logarithms use base 10 and are written as log(x). Natural logarithms use base e (approximately 2.718) and are written as ln(x).

2.3 Properties of Logarithms

Logarithms have several important properties:

log_a(1) = 0 because a⁰ = 1.
log_a(a) = 1 because a¹ = a.
log_a(xy) = log_a(x) + log_a(y).
log_a(x/y) = log_a(x) - log_a(y).
log_a(xⁿ) = n × log_a(x).

2.4 Changing Base

You can change the base of logarithms using the formula:

log_a(b) = log_c(b) / log_c(a)

for any positive number c.

2.5 Examples

Example 3: Convert 2³ = 8 to logarithmic form.
log₂(8) = 3

Example 4: Simplify log₁₀(100) using properties.
log₁₀(100) = log₁₀(10²) = 2 × log₁₀(10) = 2

3. Solving Equations

3.1 Exponential Equations

To solve an equation like 2^x = 16, you can express 16 as a power of 2:

2^x = 2⁴ → x = 4

3.2 Logarithmic Equations

For an equation like log₂(x) = 3, you convert it back to exponential form:

x = 2³ → x = 8

Watch out: Be careful with the domain of logarithmic functions. Logarithms are only defined for positive values.

Summary

Exponents can be rational and follow specific rules.
Logarithms are the inverse of exponentiation and have their own properties.
Equations can involve exponents or logarithms, and you can solve them by converting between forms.

Check your understanding

What is the value of 5⁰?
Convert the exponential equation 3^x = 81 to logarithmic form.
What is log₁₀(0.1)?
How do you express a logarithm with base 2 in terms of natural logarithms?