Quiz 3-2 (no quiz on Wed/thurs) 3. State whether the following function is growth or decay. 4. Convert into a base ‘e’ exponential function. 1. Find the constant percentage growth (decay) rate. 2. The population of “Smallville” in the year 1890 was Assume the population increased at a rate of 1.75% per year. What is the population in 1915 ?

Quiz 3-1c (Logistic Function) Initial value is 10. Limit to growth is 40. Passes thru (1, 20) c = 40 a = 3 f(0) = 10 Final equation:

3.3 Logarithmic Functions and Their Graphs

What you’ll learn about  Inverses of Exponential Functions  Common Logarithms – Base 10  Natural Logarithms – Base e  Graphs of Logarithmic Functions  Measuring Sound Using Decibels … and why Logarithmic functions are used: - in the measurement of the relative intensity of sounds - in the measurement of the relative intensity of sounds - to solve exponential growth and decay functions. - to solve exponential growth and decay functions.

What is a logarithm? A logarithm is another way of writing an exponent. Both of these equations are saying the same thing: “2 raised to what power is 8?” “2 raised to what power is 8?” The exponent is in the “exponent” position. “exponent” position. The exponent is no longer in the “exponent” position.

Exponential Logarithm Form Form basebase “base 2 to the 3 rd power is 8” power is 8” “log base 2 of 8 is 3” 3 to what power is 9 ? x = 2 x = 2 Log =

Exponential Logarithm Form Form Why did they use “b”? use “b”? Log =

Your Turn: Log = 1. Convert to logarithm form

Your Turn: Log = 6. Convert to exponential form

Vocabulary Common Logarithm: has a base of 10. We usually write it in this form: Natural Logarithm: has a base of e. We always write it in this form:

Graphs of the Common and Natural Logarithm

Base increasing: 

Your Turn: What is the base?

Evaluating Logs on your calculator log8)= Push buttons: ln 10)=

Your Turn: Use your calculator to find:

Compositions of Functions  f(2) = ? Means: wherever you see an ‘x’ in the function, replace it with a ‘2’. 1. Replace the ‘x’ with a set of parentheses. 2. Put the input value ‘2’ into the parentheses. 3. Find the output value. f(2) = 0 Cool, we found a zero of the function. zero of the function.

Function compositions Instead of a number as an input, we can use an expression as an input to a function. an expression as an input to a function.

Your turn: 17. f(g(x)) = ? 18. g(f(x)) = ?

How can you tell if functions are inverses of each other? f(x) = 0.5x + 2 g(x) = 2x - 4 IF: f(g(x)) = x and g(f(x)) = x then f(x) and g(x) are inverses of each other. then f(x) and g(x) are inverses of each other. f(g(x)) = f(2x – 4) = 0.5(2x – 4) + 2 = x g(f(x)) = g(0.5x + 2) = 2(0.5x + 2) - 4 = x

Composition of Inverse functions. If inverse functions are composed with each other, they “cancel each other out” each other, they “cancel each other out” x  f(g(x))  x x  f(g(x))  x f(x) = 0.5x + 2 g(x) = 2x - 4 f(g(2)) = 2 2  g(x)  g(2)  2(2) – 4 = 0 2  g(x)  g(2)  2(2) – 4 = 0 0  f(x)  f(0)  0.5(0) + 2  2 0  f(x)  f(0)  0.5(0) + 2  2 The input into a composition of inverse functions equals the output. equals the output.

Natural Logarithm Function

Exponential Function

and and are inverses of each other.

Exponential Function Solve for ‘x’.

Your turn: 19. Solve for ‘x’. 20. Solve for ‘x’.

Inverses of Exponential Functions Since the exponential function passes the horizontal line test; it is an even function, therefore its inverse is also a function its inverse is also a function (will pass the vertical line test). This inverse function is:

Inverses of Exponential Functions and are inverses of each other. Same thing left/right  apply the inverse of a function to “undo” that function. function to “undo” that function.

Inverses of Exponential Functions Solve for ‘x’.

Your turn 22. Solve for ‘x’. 23. Solve for ‘x’.

Finding the inverse funtion: Trade ‘x’ and ‘y’ Solve for ‘y’ The right side is a composition of inverse functions. of inverse functions.

Finding the inverse funtion: Trade ‘x’ and ‘y’ Solve for ‘y’ The right side is a composition of inverse functions. of inverse functions.

Your turn: 24. What is the inverse of: 25. What is the inverse of:

Your turn: Exponential Function Logarithmic Function

Basic Properties of Logarithms For: b > 0 and b ≠ 1 and x > 0 (y is any real number) (y is any real number) log function exponential function

Common Logarithm – Base 10 Logarithms with base 10 are called common logs If there is no base number written, it is a common log. log function exponential function

Natural Logarithm – Base ‘e’ Logarithms with base ‘e’ are called natural logs log function exponential function If it is a natural logarithm we write it differently:

Natural and Common Logarithms Your calculator uses only: common logarithms or natural logarithms

Using the Properties to Solve Equations Solve each equation by 1 st converting to exponential form: exponential form:

Evaluating a logarithm without using a calculator Remember the property These are inverses of each other They “cancel” each other, leaving as the answer: 3

Your Turn: Evaluate without using a calculator

Transforming Logarithmic Graphs Describe how to transform the graph of: Into the graph of: Shift the graph left by 2

Transforming Logarithmic Graphs Describe how to transform the graph of: Into the graph of: Reflect across the y-axis then shift the graft 2 units to the right. shift the graft 2 units to the right.

Decibels

Sound (in decibels) Level at which prolonged exposure will result in hearing damage: 90 – 96 dB result in hearing damage: 90 – 96 dB Loudest recommended level with hearing protection: 140 dB hearing protection: 140 dB How loud is a rock band on the front row ?

Your turn: (in decibels) 31. A mosquitto buzzing has a sound intensity of What is the sound level of the mosquitto?

HOMEWORK

Solving an Exponential Equation Your calculator doesn’t have base 2 (it might in some of the catalog of functions) in some of the catalog of functions) Change of Base Formula: t = 11 hours, 46 minutes