This document defines logarithmic functions and describes their properties. It defines the logarithmic function with a base a as loga x being the exponent that a must be raised to to equal x. It then lists properties of logarithms including the common logarithm with base 10 and the natural logarithm with base e. The document provides the basic definitions and properties of logarithmic functions.

1. 4.3 Logarithmic
Functions

2. Definition of the Logarithmic
Function pg. 344
Let 𝑎 be a positive number with 𝑎 ≠ 1. The
logarithmic function with base 𝒂, denoted
log𝑎, is defined by
log𝑎 𝑥 = 𝑦 ↔ 𝑎𝑦
= 𝑥
So log𝑎 𝑥 is the exponent to which the base
𝑎 must be raised to give 𝑥

3. Properties of Logarithms pg.
345
1) log𝑎 1 = 0 We must raise 𝑎 to the
power 0 to get 1
2) log𝑎 𝑎 = 1 We must raise 𝑎 to the
power 1 to get 𝑎
3) log𝑎 𝑎𝑥
= 𝑥 We must raise 𝑎 to the
power 𝑥 to get 𝑎𝑥
4) 𝑎log𝑎 𝑥
= 𝑥 log𝑎 𝑥 is the power to which
𝑎 must be raised to get 𝑥

4. Common Logarithm pg. 348
The logarithm with base 10 is called the
common logarithm and is denoted by
omitting the base
log 𝑥 = log10 𝑥

5. Natural Logarithm pg. 349
The logarithm with base 𝑒 is called the
natural logarithm and is denoted by ln
ln 𝑥 = log𝑒 𝑥
So we have
ln 𝑥 = 𝑦 ↔ 𝑒𝑦
= 𝑥

6. Properties of Natural
Logarithm pg. 349
1) ln 1 = 0 We must raise 𝑒 to the
power 0 to get 1
2) ln 𝑒 = 1 We must raise 𝑒 to the
power 1 to get 𝑒
3) ln 𝑒𝑥
= 𝑥 We must raise 𝑒 to the
power 𝑥 to get 𝑒𝑥
4) 𝑒ln 𝑥
= 𝑥 ln 𝑥 is the power to which 𝑒
must be raised to get 𝑥