6.3A – Logarithms and Logarithmic Functions

Objective: TSW evaluate logarithmic expressions.

Definition of a Logarithmic Function

For x > 0 and b > 0,

y = logb x is equivalent to by = x.

So, a logarithm allows us to solve for unknown exponents since a logarithm is an exponential equation

Logarithmic form: y = logb x Exponential Form: by = x. Logarithmic form: y = logb x Exponential Form: by = x.

Examples….

1. Write each equation in its equivalent exponential form.a. 2 = log5 x b. 3 = logb 64 c. log3 7 = y

We will use the fact that: y = logb x means by = x,

Examples…

2. Write each equation in it’s equivalent logarithmic form.

a. 2r = 10 b. 17= b3.2 c. 122 = 144

Now Let’s Apply the Definition to Solve.Here are the steps:

1. Set the logarithm equal to y

2. Rewrite the logarithm in exponential form

3. Break the bases down so they are the same!!!

4. REMEMBER, if the bases are equal, the exponents MUST BE equal!

5. Set the exponents equal and solve for y.

6. CHECK YOUR SOLUTION!

Logarithmic form: y = logb x Exponential Form: by = x. Logarithmic form: y = logb x Exponential Form: by = x.

Let’s Try a Few…

3. Evaluate

a. log2 16 b. log3 9

4. Evaluate

a. log5(1/25) b. log25 5

Homework!

Pg. 496 #’s 1-7(all), 13-23(odds), 25-36(all).