Warm ups 1. Write the equation in exponential form. 2. Solve the equation e^x=13.7 x=7

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.

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.

Example Find the value of log8172 using the change of base formula. Using base 10 allows you to put it into the calculator.

Example Given that log 5 = 0.6990, evaluate each expression. a. log 50,000 b. 4 is the characteristic 0.6990 is the mantissa

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.

Example Solve

Example Solve: a. 54x = 73 (take the log of both sides) b. 2.2x-5 = 9.32 x=.6665 x=7.8311

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

pH What would be the hydrogen ion concentration of vinegar with pH = 3? H+=.001

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

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) 794,328