Logarithms and Exponential Functions

Learning Objectives

Define a logarithm as the inverse of an exponential function.

Convert between logarithmic and exponential forms.

Use key properties of logarithms (product, quotient, and power rules) to simplify expressions.

Solve basic logarithmic and exponential equations.

The Inverse Relationship

Exponential functions and logarithmic functions are inverses of each other.

An exponential function is of the form y = bˣ, where b is the base and x is the exponent. It models rapid growth or decay.

A logarithmic function is of the form y = logₐ(x). It answers the question: 'What exponent do I need to raise the base b to in order to get x?'

The relationship is:

y = bˣ <=> logₐ(y) = x

Example:

Exponential form: 2³ = 8

Logarithmic form: log₂(8) = 3 (Read as: 'log base 2 of 8 is 3')

A common logarithm has a base of 10 (log₁₀(x) is often written as log(x)).

A natural logarithm has a base of e (Euler's number, ≈ 2.718) and is written as ln(x).

Properties of Logarithms

These rules are essential for simplifying expressions and solving equations.

1.Product Rule: logₐ(xy) = logₐ(x) + logₐ(y)

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

2.Quotient Rule: logₐ(x/y) = logₐ(x) - logₐ(y)

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

3.Power Rule: logₐ(xⁿ) = p * logₐ(x)

(The log of a number raised to a power is the power times the log of the number).

4.Change of Base Formula: logₐ(x) = log(x) / log(b)

(Allows you to calculate a log of any base using a calculator that only has log and ln).

Solving Equations

To solve an exponential equation where the variable is in the exponent (e.g., 3ˣ = 15), take the logarithm of both sides.

log(3ˣ) = log(15)

x * log(3) = log(15)

x = log(15) / log(3)

To solve a logarithmic equation (e.g., log₂(x) = 5), convert it to exponential form.

2⁵ = x

x = 32

Key Terms