 Logarithmic Expressions MathBitsNotebook.com Terms of Use   Contact Person: Donna Roberts A logarithm is an exponent.
In the example shown at the right, 3 is the exponent to which the
base 2 must be raised to create the answer of 8, or 23 = 8.
In this example, 8 is called the antilogarithm base 2 of 3. Try to remember the "spiral" relationship between the values as shown at the right. Follow the arrows starting with base 2 to get the equivalent exponential form, 23 = 8.  A logarithm base b of a positive number x is such that: for b > 0, b≠ 1, logb x = y if and only if by = x. The log bx is read "log base b of x". The logarithm y is the exponent to which b must be raised to get x.

 Logarithms with base 10 are called common logarithms. When the base is not indicated, base 10 is implied. Logarithms with base e are called natural logarithms. Natural logarithms are denoted by ln.    Logarithms with base 10 are called common logarithms and are written without the 10 showing.
 The log key will calculate common (base 10) logarithms. Logarithms with the base e are called natural logarithms and are written using the notation ln(x).
 The ln key will calculate natural (base e) logarithms. For "other bases" use the change of base formula:  is entered as Origins of Change of Base Formula: Set = x. Convert to exponential form. Take common log of both sides. User power rule. Divide by log b.
 Change of Base Formula: Calculator Entry:   The
logBASE( operation template can also be used.
To load the template go to
MATH → arrow down to A: logBASE(.
For more options, see link below: For more help with logarithms on your calculator, click here. Properties of Logarithms
Using the properties of exponents, we can arrive at the properties of logarithms.
 Properties of Exponents: Let's find the connection! Similar investigations lead to the other logarithm properties.
 Properties of Logarithms:   Also, These log properties remain the same when
working with the natural log. Remember:    ln 1 = 0    and    ln e = 1  Examples:

 1 Write in exponential form. Solution: 2 Write in logarithmic form. Solution: 3 Evaluate: Solution: If using your calculator, remember to use the change of base formula and enter log 1 / log 3. 4 What is the value of x? Solution: 5 Write in expanded form: (Apply the "properties of logs" rules.) Solution: 6 Write in expanded form: Solution: 7 Express as a single logarithm: (Apply the properties of logs rules in reverse) Solution: 8 Express as a single logarithm: Solution: 9 Using properties of logarithms, show that Solution: 10 Using properties of logs, solve for x. Solution:  