Warm ups

• 1. Write the equation in exponential form.

• 2. Solve the equation

x7.13ln

70log10loglog 222 x

Lesson 11-5 Common Logarithms

Objective: To find common logarithms and antilogarithms of numbers

To solve equations and inequalities using common logarithms

To solve real-world applications with common logarithmic functions

Common Logarithms

Logarithms with base 10 are common logarithms. This is what your calculator uses to find logarithms.Common logarithms are made up of 2 parts: the characteristic and the mantissa. In the equation the mantissa is the number between 0 and 1 so it would be .8451 or log 7. The characteristic is the exponent of ten when the number is written in scientific notation. The log is expressed as the sum of the mantissa and characteristic.

8451.7log

Common Logs

• The common or base-10 logarithm of a number is the power to which 10 must be raised to give the number.

• Since 100 = 102, the logarithm of 100 is equal to 2. This is written as:

Log(100) = 2.• 1,000,000 = 106 (one million), and

Log (1,000,000) = 6.

Logs of small numbers

• 0.0001 = 10-4, and Log(0.0001) = -4.All numbers less than one have negative logarithms.

• As the numbers get smaller and smaller, their logs approach negative infinity.

• The logarithm is not defined for negative numbers.

Change of Base Formula

If a, b and n are positive numbers and neither a nor b is 1, then the following equation is true.

loglog

logb

ab

nn

a

Example

Find the value of log8172 using the change of base formula.

8log

172log Using base 10 allows

you to put it into the calculator.

4754.29031.

2355.2

ExampleGiven that log 5 = 0.6990, evaluate each expression.a. log 50,000

b. 313

log7

)5000,10log( 5log10log 4

6990.04 4 is the characteristic0.6990 is the mantissa

6990.4

7log13log3 8451.03418.3

4967.2

Antilogs

• The operation that is the logical reverse of taking a logarithm is called taking the antilogarithm of a number. The antilog of a number is the result obtained when you raise 10 to that number.

• The antilog of 2 is 100 because 102=100.• The antilog of -4 is 0.0001 because 10-4 = 0.0001

Make sure you can use your calculator to generate this table.

N As a power of 10 Antilog(N)

3 103 1000

1.5 101.5 31.62

1 101 10

0 100 1

-2 10-2 0.01

-3.4 10-3.4 0.0003981

Example

Solve 16.7 log

x

xantianti

1loglog7.6log

xanti

17.6log

7.6log

1

antix

7.610

1x 7109953.1 x

ExampleSolve:a. 54x = 73 (take the log of both sides)

b. 2.2x-5 = 9.32

Applications of Logarithms

• Logarithms are used in real world applications including pH and the Richter scale (earthquakes).

pH• pH defined as pH =

• where [H+] is hydrogen ion concentration– measured in moles per liter

• ex: pH of 6.7 is solved the same way our previous equation

H

1log

16.7 log

x

pH

• What would be the hydrogen ion concentration of vinegar with pH = 3?

H

1log3

Earthquake – Richter scale

• R = log It compares how much

• stronger the earthquake is compared to a given standard

• R= 3.0 then 3 = log 1000 = I = 1000I0 1000 times the standard

0I

I

0I

I

0I

I

Earthquake – Richter scale• Haiti 7.0 7 = log

• 10,000,000 =

• Japan 8.9 8.9 = log

• 794,328,235 =

• Virginia 5.9 ?(August 23, 2011)

0I

I

0I

I

0I

I

0I

I

794,328