Common Logarithmic Functions

A logarithmic function is the inverse to an exponential function. Just like exponential function have common bases and a natural base; logarithmic functions have common logs and a natural log.

This discussion will focus on the common logarithmic functions.

The general common logarithmic equation is:

COMMON LOGARITHMIC FUNCTION


y=lo g a  x      if and only if     x = ay

Where a > 0, a ≠ 1 and x > 0



When reading lo g a  x say, "log base a of x".

Some examples are:

1. lo g 10  100=2 because 102 = 100

2. lo g 3  81=4 because 34 = 81

3. lo g 15  225=2 because 152 = 225

Notice in the examples that the base of the log is also the base of the corresponding exponent. In example 1 above, the logarithmic function has a log of base 10 and the corresponding exponential function has a base of 10.

If you see log with no base is means log of base 10 or log = log10.

Some basic properties of logarithmic functions are:

Property 1: lo g a  1=0 because a0 = 1

Property 2: lo g a  a=1 because a1 = a

Property 3: If lo g a x=lo g a y , then x = y       One-to One Property

Property 4: lo g a   a x =x and a log a x =x            Inverse Property


Let's solve some simple logarithmic equations:

log x = 4

Step 1: Choose the most appropriate property.


Properties 1 and 2 do not apply, as the log equals neither 0 nor 1. Property 3 does not apply since a log is not set equal to a log of the same base. Therefore Property 4 is the most appropriate.

Property 4 - Inverse

Step 2: Apply the Property.


Remember log=lo g 10 . Since the log has a base of 10, taking the inverse means to rewrite both sides as exponents with base 10.

log x = 4 Original


10logx = 104 Exponent of 10

Step 3: Solve for x.

Property 4 states that a lo g a x =x , therefore the left-hand side becomes x.

x = 104   Apply Property


x = 10,000 Evaluate

Example 1:      lo g 3  x=lo g 3  4x9

Step 1: Choose the most appropriate property.


Properties 1 and 2 do not apply, as the log equals neither 0 nor 1. Since a log is set equal to a log of the same base. Property 3 is the most appropriate.

Property 3 - One to One

Step 2: Apply the Property.


Property 3 states that if lo g a x=lo g a y , then x = y. Therefore x = 4x - 9.

x = 4x - 9 Apply Property

Step 3: Solve for x.

-3x = -9    Subtract 4x


x = 3        Divide by -3

Example 2:      lo g 3  3x=5

Step 1: Choose the most appropriate property.


Properties 1 and 2 do not apply, as the log equals neither 0 nor 1. Property 3 does not apply since a log is not set equal to a log of the same base. Therefore Property 4 is the most appropriate.

Property 4 - Inverse

Step 2: Apply the Property.


Since the log has a base of 3, taking the inverse means to rewrite both sides as exponents with base 3.

lo g 3  3x=5 Original


3 log 3 3x = 3 5 Exponent of 3

Step 3: Solve for x.


Property 4 states that a lo g a x =x , therefore the left-hand side becomes x.

3x = 35       Apply Property


x= 243 3      Divide by 3


x = 81        Evaluate





Related Links:
Math
algebra
Natural Base - Simple Equations
Completing the Square when a = 1
Algebra Topics


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