Common Logarithmic Functions
This discussion will focus on the common logarithmic functions.
The general common logarithmic equation is:
COMMON LOGARITHMIC FUNCTION
if and only if x = ay
Where a > 0, a ≠ 1 and x > 0
When reading say, "log base a of x".
Some examples are:
1. because 102 = 100
2. because 34 = 81
3. 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 2: because a1 = a
Property 3: If , then x = y One-to One Property
Property 4: and Inverse Property
Let's solve some simple logarithmic equations:
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 . 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 , therefore the left-hand side becomes x. |
x = 104 Apply Property x = 10,000 Evaluate |
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 , 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 |
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. |
Original Exponent of 3 |
Step 3: Solve for x. Property 4 states that , therefore the left-hand side becomes x. |
3x = 35 Apply Property 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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