March 2003S. H. Lapinski Logarithms Tutorial Understanding the Log Function

Where Did Logs Come From? The invention of logs in the early 1600s fueled the scientific revolution. Back then scientists, astronomers especially, used to spend huge amounts of time crunching numbers on paper. By cutting the time they spent doing arithmetic, logarithms effectively gave them a longer productive life.

There are still good reasons for studying them. To model many natural processes, particularly in living systems. We perceive loudness of sound as the logarithm of the actual sound intensity, and dB (decibels) are a logarithmic scale. To measure the pH or acidity of a chemical solution. To measure earthquake intensity on the Richter scale.

How they are developed In the mathematical operation of addition we take two numbers and join them to create a third = 8 We can repeat this operation: = 12 Multiplication is the mathematical operation that extends this: 3 4 = 12 In the same way, we can repeat multiplication: = 27 The extension of multiplication is exponentiation: = 27 = 3 3

More on development Now consider that we have a number and we want to know how many 2's must be multiplied together to get that number. For example, given that we are using `2' as the base, how many 2's must be multiplied together to get 32? We want to solve this equation: 2 B = = 32, so B = 5. mathematicians made up a new function called the logarithm: log 2 32 = 5

DEFINITION: a logarithm function y = log a x is the inverse of a exponential function y = ab x If you want to undo a exponent, use a logarithm

Using Common Log to solve exponential equations What power to I raise 3 to, to get 27? What power to I raise 4 to, to get 16? What power to I raise 2 to, to get 32? What power to I raise 6 to, to get 1,296? As you can see, these get difficult…. We can use common logs to solve them

Using Common Log to solve exponential equations we will start with one we know, write down the steps 1. Isolate the base with the exponent 2. Rewrite it in log notation a. move the exponent out front b. Write log in front of the two bases 3.Solve for x, divide both sides by by everything but the x 4.Find the log button on the calculator (it is next to the 7) 5. log(27) / log(3) enter

Using Common Log to solve exponential equations Now, a harder one… Rewrite it in log notation a. move the exponent out front b. Write log in front of the two bases 3.Solve for x, divide both sides by log 6 4.Find the log button on the calculator (it is next to the 7) 5. log(1296) / log(6) enter 1. Isolate the base with the exponent

Using Common Log to solve exponential equations Now, a harder one… Rewrite it in log notation a. move the exponent out front b. Write log in front of the two bases 3.Solve for x, divide both sides by 4log 3 4. log(6561) /(4 log(6) enter 1. Isolate the base with the exponent