Introduction to Logarithmic Functions

In this lesson, we will look at what are logarithms and the relationship between exponents and logarithms.

Logarithms can be considered as the inverse of exponents (or indices).

Definition of Logarithm

If a^x = y such that a > 0, a ≠ 1 then log_a y = x

a^x = y ↔ log_a y = x

Exponential Form

y = a^x

Logarithmic Form

log_a y = x

Remember: The logarithm is the exponent.

Example:

Convert the following exponential form to the logarithmic form:

a) 4² = 16

b) 2⁵ = 32

c)

Solution:

a) 4² = 16
2 = log₄ 16 (the log is the exponent)

b) 2⁵ = 32
5 = log₂ 32

Example:

Convert the following logarithmic form to exponential form

a) 3 = log₂ 8

b) 2 = log₅ 25

c)

Solution:

a) 3 = log₂ 8
2³ = 8

b) 2 = log₅ 25
5² = 25

Take note of the following:

• Since a¹ = a, log_a a = 1
• Since a⁰ = 1, log_a 1 = 0
• Log_a 0 is undefined
• Logarithms of negative numbers are undefined.
• The base of logarithms can be any positive number except 1.
• Logarithms to the base 10 are known as common logarithms and are represented by log₁₀ or log.
• Logarithms to the base e are known as natural logarithms and are represented by log_e or ln.

Videos

An introduction to logarithmic functions -
Professor Edward Burger introduces logarithmic functions

Converting between exponential and logarithmic forms -
Professor Edward Burger explains converting between exponential and logarithmic functions

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