Algebra 2/Trigonometry Name: __________________________ Section 8.4 – Logarithms Notes Date: ___________________________ Recall: To find an inverse of a function, _______________________________________. Create a table of values for y = 2x. Use the table of values to graph y and its inverse. x y x y This inverse is called a ____________________________________________________ ______________________________________________________________________ Definition of Logarithmic Function with Base a– For x > 0, a > 0, and a ¹ 1, Graphing Logarithmic Functions Characteristics: Shifting Graphs of Logarithmic Functions

__________________________________________ Converting Logarithms to Exponentials Ex.1: Ex.2: Ex.3: Ex.4: Converting Exponentials to Logarithms Ex.1: Ex.2: Ex.3: Ex.4: The logarithmic function _________________________________________ ________________________________________________________________________________________________________________________ __________________________________________ ____________________________________________________________ Evaluating Common Logarithms. Given f(x) = logx, find: Example 1: Example 2: Example 3: Evaluating Logarithms (other than base 10) Example 1: Example 3: Example 2: Example 4:

Algebra 2/Trigonometry Section 8.4 – Logarithms Notes – Day 1 Properties of Logarithms: 1) ____________________________________________________________________ 2) ____________________________________________________________________ 3) ____________________________________________________________________ 4) ____________________________________________________________________ Using Properties of Logarithms: Example 1: Example 2: Example 3: Example 4: Example 5: Example 6:

Algebra 2/Trigonometry Section 8.4 - Natural Base & Natural Logarithms Notes – Day 2 The Natural Base, e – In many applications, the natural base, e, is used. e » 2.718281828… is an ____________________________________________. The function given by f(x) = ex is called the ____________________________________. e is the constant term and x is the variable. Evaluating the Natural Exponential Function: Graph of the Natural Exponential Function: The Natural Logarithmic Function – Evaluating the Natural Logarithmic Function: Properties of Natural Logarithms: 1) ____________________________________________________________________ 2) ____________________________________________________________________ 3) ____________________________________________________________________ 4) ____________________________________________________________________

Using Properties of Natural Logarithms: Example 1: Example 2: Example 3: Example 4: Example 5: Example 6: Example 7: Compounding Interest Continued x 1 10 100 1,000 10,000 100,000 1,000,000  What if the interest compounded continuously? Compounded Continuously Interest Formula Example - A total of $12,000 is invested at an annual interest rate of 9%. Find the balance after 5 years if it is compounded… quarterly. (b) monthly. (c) continuously.

Population Growth/Decay P = Population C = Initial value k = rate of growth/decay t = number of years The population of a small town continuously grows at a rate of 0.5%. If the population is currently about 4,300 people ... Write an equation that models the situation b) Predict the population next year c) Predict the population 10 years from now Example - You purchase a car for $40,000, and it depreciates in value at a rate of 11%. a) Write an equation to represent the situation. b) Find the approximate value of your car after 8 years. Class work: Working with your partner, complete the table (on a separate piece of paper) using the given problem situation. Show all work and round any answers to the nearest hundredth (nearest cent). You will be turning this in!! You are adding compounding continuously. If you made any mistakes for your last submission, please fix them for this one! Compounded Interest Earned Value of Account Annually Monthly Weekly Daily Hourly Continuously Imagine you were given $10,000 for graduation from high school. If you invested that amount into an account that yields an APR of 4.5% and left the money in your investment for the next 10 years, how much money would you have? How much interest would you accrue?