Logarithmic Functions The inverse of the equation y = b x is x = b y Since there is no algebraic method for solving x = by by for y in terms of x,x, the Logarithmic Function is used to allow y to be expressed in terms of x.x. That’s right! Interchange x and y. Sounds pretty easy so far. Let’s move on.

Let’s Take a Closer Look at Some Logs A logarithm is really an exponent written in a different form. The equation y = b x is an exponential function Let’s break this down. b is the base x is the exponent y is the value of b x Now let’s bring in the logs. Written in logarithmic form, the equation y = b x would be x = log b a We read this as x is the logarithm of a with base b

Breaking Down Logs Let’s look at a log piece by piece. The equation x = log b a is a logarithmic function Let’s break this down. b is the base x is the exponent a is the value of b x Hey! I’ve seen this before. It’s Sam Ting as breaking down exponential functions. That was easy

Comparing Logarithmic form and exponential form Exponential FormLogarithmic Form y = b x x = log b a 32 = = log = = log = log = log = = 5 3 Asi De Facil

Logarithms with Variables 3 = log 4 a In each equation, find the value of the variable since 4 3 = 64, a = 64 x = log 6 36 since 6 2 = 36, x = 2 3 = log b 125 since 5 3 = 125, b = 5 Hey, I can just use my calculator for this. This looks a little harder. Maybe I should use a real calculator for this one. That was easy 43 = a43 = a6 x = 36b 3 = 125

More Logarithms with Variables In each equation, find the value of the variable 5 = log 8 a since 8 5 = 32,768, a = 32,768 x = log 7 2,401 since 7 4 = 2,401, x = 4 3 = log b 6,859 since 19 3 = 6,859, b = 19 Hey, those are some pretty big numbers. I hope my calculator knows how to do this. That was easy 8 5 = a7 x = 2,401b 3 = 6,859

Common Logs Any logarithm with base 10 is a Common Log When writing a common logarithm, the base is usually omitted. So, 5 = log ,000 and 5 = log 100,000 are Sam Ting. Let’s compare Logarithmic Form and Exponential Form of some Common Logs. Exponential FormLogarithmic Form 3 = log 1,0001,000 = ,000,000 = = log 1,000,000 10,000 = = log 10,000 That was easy

Common Logs with Variables In each equation, find the value of the variable x = log x = 100 since 10 2 = 100, then x = 2 count the zeros 7 = log a 10 7 = a since 10 7 = 10,000,000, then a = 10,000,000 write the proper number of zeros Hey, I don’t even need a calculator for this! That was easy

More Common Logs with Variables Find the value of the variable to the nearest one hundredth x = log 1, x = 1,345 Hey, there’s no zeros to count = log a = a That was easy We could use the LOG key on our calculator. LOG (1,345) = 3.13 What’s the proper number of zeros? We could use the 10 x key on our calculator =

Change of Base How can I get my calculator to evaluate logs in bases other than base 10? That’s easy, just use the Change of Base Formula x = log 8 512= 3 x = log ,832= 5 It’s time to push the easy button once again!

More Change of Base Let’s throw some decimals into the mix. x = log 4 32 x = log = 3 = 2.5 This stuff is too easy. Soon I’ll have to buy new batteries for my easy button. x = log , = That was easy