Intro to
Logarithms
Definition
A logarithm base b of a positive number
x satisfies the following definition.
For 𝑏 > 0, 𝑏 ≠ 1, 𝑙𝑜𝑔𝑏 𝑥 = 𝑦 if and only if
𝑏 𝑦 = 𝑥.
You can read 𝑙𝑜𝑔𝑏 𝑥 as “log base b of x.”
In other words, the logarithm y is the
exponent to which b must be raised to
get x.
Different Bases
Logs may be written three different ways,
depending on the given base:
= 𝑦: used for a log with a given
base #
 𝑙𝑜𝑔𝑥 = 𝑦: called the common log and
used for base 10 (no base showing)
 𝑙𝑛𝑥 = 𝑦: called the natural log and used
for base e (spelled differently)
 𝑙𝑜𝑔𝑏 𝑥
Exponent to Log form.
 Exponential Form:
𝑦 = 𝑏𝑥, 𝑏 > 0
Log Form:
𝑙𝑜𝑔𝑏 𝑦 = 𝑥 , 𝑏 > 0
Change from log to exponential form
log 81  0
log 1000  3
1
log 2    4
 16 
Changing Forms
Change from exponential to log form
2x  8
y 1  4
1
2
9 3
Evaluate
log 2 64
5
log 4 n 
2
Solving log equations by
switching forms
log 2 x  3
log 27 3  x
Solving log equations by
switching forms
1
log x 4 
2
1
log 5
x
25
Log is the INVERSE of Exponential
Characteristics of Inverses
1. Swap domain and range
2. Graphs are reflections across the line
𝑦=𝑥
Consider:
𝑦 = 2𝑥
and
𝑦 = 𝑙𝑜𝑔2 𝑥
Look at the graphs
𝑦 = 2𝑥
𝑦 = 𝑙𝑜𝑔2 𝑥
Parent Graph of the Log Function
y  log(x)
Graph of a Log function
y  log( x  1)
Graph of a Log function
y  log( x)  2
Graph of a Log function
y  log( x  3)  1
Graph of a Log function
y   log(x)
Graph of a Log function
y  log( x)
Graph of a Log function
y   log( x)  3