Real Exponents

A common logarithm is made up of two parts, the characteristic and the
mantissa. The mantissa is the logarithm of a number between 1 and 10. Thus, the
mantissa is greater than 0 and less than 1.
In Example 1, the mantissa is log 7 or 0.8451. The characteristic is the
exponent of ten that is used to write the number in scientific notation. So, in
Example 1, the characteristic of 1,000,000 is 6, and the characteristic of 0.0001 is
4. Traditionally, a logarithm is expressed as the indicated sum of the mantissa
and the characteristic.
You can use a calculator to solve certain equations containing common
logarithms.
Example 2
Real World
A p plic atio n
Graphing
Calculator
Tip
The key marked
LOG on your
calculator will
display the common
logarithm of a number.
CHEMISTRY Refer to the application at the beginning of the lesson. If the
water being tested contains 7.94 10 9 moles of H per liter, what is the pH
level of the water?
pH log H
1
1
pH log H 7.94 10 9
7.94
10 9
Evaluate with a calculator.
LOG 1 7.94 2nd [EE] (–) 9 ENTER
pH 8.1
The pH level of the water is about 8.1
The properties of logarithmic functions you learned in Lesson 11-4 apply to
common logarithms. You can use these properties to evaluate logarithmic
expressions.
Example 3
Evaluate each expression.
a. log 5(2) 3
log 5(2) 3 log 5 3 log 2
0.6990 3(0.3010)
0.6990 0.9031
1.6021
Product Property, Power Property
Use a calculator.
b. log 1 9
2
6
log 1 9
2 2 log 19 log 6 Quotient Property, Power Property
6
2(1.2788) 0.7782 Use a calculator.
2.5576 0.7782
1.7794
Lesson 11-5 Common Logarithms 727