Warm Up Find the algebraic inverse: 𝑓 𝑥 = 15𝑥 – 1 𝑓 𝑥 = 𝑥−2 2 𝑓 𝑥 = 𝑥−2 2 𝑓 𝑥 = 1 3 𝑥+7 𝑓 𝑥 = 𝑥−4 𝑓(𝑥) = −5𝑥−11

Answers Find the algebraic inverse: 𝑓 𝑥 = 15𝑥 – 1 𝑓(𝑥) −1 = 𝑥+1 15 𝑓 𝑥 = 15𝑥 – 1 𝑓(𝑥) −1 = 𝑥+1 15 𝑓 𝑥 = 𝑥−2 2 𝑓(𝑥) −1 = 𝑥 +2 𝑓 𝑥 = 1 3 𝑥+7 𝑓(𝑥) −1 =3𝑥−21 𝑓 𝑥 = 𝑥−4 𝑓(𝑥) −1 = 𝑥 2 +4 𝑓 𝑥 = −5𝑥−11 𝑓(𝑥) −1 = −𝑥−11 5

Common Logs

Common Logs Investigation In your groups, work on problems 1 and 2 on the handout. Resource Manager: Obtain a worksheet for each person Time Keeper: 10 minutes Reader/Recorder: Read each question out loud Spy Monitor: Check in with other groups or my key

Common Logs Investigation Express each of the numbers below as accurately as possible as a power of 10 (i.e. y=10x). You can find exact values of the required exponents by thinking about the meanings of positive and negative exponents.   100 10,000 0.0001 0.01 102 104 10-4 10-2

Logarithms As you may have noticed, it is not easy to solve equations like 10x = 9.5 or 10x = 0.0023, even by estimation. To deal with this problem, mathematicians have developed a procedure for finding missing exponents! Remember working with inverses? Exponential functions have an inverse operation that allows you to find a missing exponent. This type of function is called a logarithm.

We define a logarithm as follows: If y = bx, then logb y = x.

Logarithms In this class, we will focus on logarithmic problems in base 10 (b = 10), called common logarithms. Log10 a is pronounced “log base 10 of a”. Because base 10 logarithms are so commonly used, log10 a is often written as log a (we assume the b=10 just like we assume the index of 2 in ). Most calculators have a built-in log function that automatically finds the required exponent value.

Common Logs In your groups, work on problems 3 – 5 on the handout. Time Keeper: 10 minutes Reader/Recorder: Read each question out loud Spy Monitor: Check in with other groups or my key

Homework Independent Practice on Common Logarithms