Integers as Exponents Simplify:

Rational Numbers as Exponents Definition of : Definition of : What about: ?

Rational Numbers as Exponents If p and q are integers, with q>0, and b is a positive real number, then:

Rational Numbers as Exponents Simplify:

Rational Numbers as Exponents All properties of exponents we’ve already learned apply to rational exponents as well. Ex) Re-write as a product of powers:

Rational Numbers as Exponents All properties of exponents we’ve already learned apply to rational exponents as well. Ex) Simplify:

Rational Numbers as Exponents Ex) Write in exponential form and simplest radical form:

Learning Log Summary LT 1 – I can simplify expressions involving rational exponents. A rational exponent means… To multiply bases with rational exponents…

Solving Equations with Rational Exponents Ex) Solve:

Solving Equations with Rational Exponents Ex) Solve:

Learning Log Summary LT 2 – I can solve equations involving rational exponents. The goal of solving an equation is… To “undo” a rational exponent on a variable…

Closure Homework: Pg. 458 ~ 1-35 (Odd), 43, 45

Paper-Folding Using a piece of scrap paper, fold the paper to determine how many rectangles exist (only single rectangles) after each fold. Record data in a table.

The Exponential Function We learned about an expression like yesterday. It is also possible that the exponent is an irrational number, like:

The Exponential Function Consider where x can be any rational or irrational number. x y Where is the value of ?

Extending the Laws of Exponents to Irrationals Ex) Simplify:

The Exponential Function If b>0, the function defined by is called the exponential function with base b. We will explore the graph of this function on Thursday!

Learning Log Summary LT 3 – I can define an exponential function and describe the meaning of the variables in the equation . An exponential function is one where… The base of an exponential function is…

Solving Exponential Equations An exponential equation is an equation in which a variable appears in the exponent position. A simple example:

Solving Exponential Equations Solve: Express both sides using the same base. Set the exponents equal and solve for ‘x’. Check your solution.

Solving Exponential Equations Solve: Express both sides using the same base. Set the exponents equal and solve for ‘x’. Check your solution.

Learning Log Summary LT 4 – I can solve exponential equations by re-writing necessary bases. An exponential equation is… To solve an exponential equation…

Closure Homework: Pg. 461 ~ 1-29 (Odd)

Exponential Function Transformations Activity Using desmos.com as a graphing tool, graph the examples listed in problems 1 and 2 and answer the questions based on the insights gained from the graphs.

Learning Log Summary Revisited LT 3 – I can define an exponential function and describe the meaning of the variables in the equation . The exponential function can be transformed by… The effect that b has on the exponential function is…

Function Composition Function composition is a way of performing two functions in one. Given two functions f(x) and g(x), f(g(x)) means “First apply g(x), then apply f(x).” This is called the composite of the two functions. f(g(x)) is read aloud as “f of g”.

Function Composition

Function Composition

Function Composition

Learning Log Summary LT 5 – I can compose two functions and simplify the result. To compose two functions… f(g(x)) means…

Inverse Functions

Inverse Functions How can we verify that f and g “undo” each other for every possible x value?

Inverse Functions The functions f and g are inverse functions if: for all x in the domain of g and for all x in the domain of f. The inverse of function f is written f -1.

Learning Log Summary LT 6 – I can find the inverse of a function and determine if two functions are inverses using function composition. To find the inverse of a function… If two functions are inverses…

Inverse Functions

Inverse Functions

Learning Log Summary LT 7 – I can describe the graphs of inverse functions and use the Horizontal Line Test appropriately. The graphs of inverse functions… If a functions passes the Horizontal Line Test…

Closure Homework: Pg. 466 ~ 1-14 (All)

The Logarithmic Function Does the function have an inverse function? Why? How can we find it?

The Logarithmic Function For any point (x,y) on the graph of the exponential function…

Learning Log Summary LT 8 – I can describe the logarithmic function and explain its relationship to the exponential function. The logarithmic function is… The connection between an exponential and logarithmic function is…

The Logarithmic Function “log base 2 of a” If , then means… means… means… means… means…

The Logarithmic Function Definition of a Logarithm: If b and N are positive numbers, : if and only if

The Logarithmic Function Write each equation in exponential form:

The Logarithmic Function Write each equation in logarithmic form:

The Logarithmic Function Simplify each logarithm:

The Logarithmic Function Solve each equation:

Learning Log Summary LT 9 – I can write exponential equations in logarithmic form (and vice-versa) and solve logarithmic equations. To solve a logarithmic equation… To re-write a logarithm expression as an exponential expression…

Closure Homework: Pg. 470 ~ 1-25 (Odd)

LT 1-7 Review Simplify:

LT 1-7 Review Simplify:

LT 1-7 Review Simplify:

LT 1-7 Review Simplify:

LT 1-7 Review Solve:

LT 1-7 Review Solve:

LT 1-7 Review If and find .

LT 1-7 Review If and find .

LT 1-7 Review If and find .

LT 1-7 Review Determine if the functions are inverses:

LT 1-7 Review Find if .

Properties of Logarithms Let b be the base of a logarithmic function and let M and N be positive numbers.

Properties of Logarithms Determine if the statement if true or false.

Properties of Logarithms Ex) Express in terms of and .

Properties of Logarithms Ex) Express in terms of and .

Properties of Logarithms Ex) Express as a single logarithm:

Properties of Logarithms Ex) If and , find:

Properties of Logarithms Ex) If and , find:

Properties of Logarithms Ex) Solve: Check your solutions!

Learning Log Summary LT 10 – I can utilize the properties of logarithms to re-write and evaluate logarithmic expressions. The properties of logarithms are… To re-write a logarithmic expression…

Closure Homework: Pg. 476 ~ 1-25 (Odd)

Solving Exponential Equations Because of its simplicity, the expression is called the common logarithm. It is often written as just .

Solving Exponential Equations Ex) Simplify: Between what two integers must x fall?

Solving Exponential Equations Ex) Determine between which two integers x must fall. Then solve for x:

Solving Exponential Equations Using the Calculator to find Roots:

Solving Exponential Equations Solve:

Solving Exponential Equations Solve: “Calculation Ready” / Exact vs. Approximate Value

Solving Exponential Equations Determine between what two integers t must fall, then find the approximate value of t.

Solving Equations Note the difference between the equations:

The Change of Base Formula Simplify:

The Change of Base Formula

Learning Log Summary LT 11 – I can solve exponential equations by using logarithms and use the Change-of-Base Formula to evaluate a logarithm of any base. To solve an exponential equation… The Change of Base Formula is…

Closure Homework: Pg. 481 ~ 1-37 (Odd)

Exponential Growth and Decay Suppose you invest “P” dollars in an account that earns 8% interest, compounded annually… Value = (Amt at Beginning of Year) + Interest Earned = 1.00(Amt at Beginning of Year) + .08(Amt at Beginning of Year) = 1.08(Amt at Beginning of Year) Time (years) 1 2 3 … t Value P

Exponential Growth and Decay How long will it take an investment of $1000 to triple in value if it is invested at an annual rate of 12%, compounded quarterly?

Exponential Growth and Decay A population grows by 50% every year. Time (years) 1 2 3 … t Value P A population decreases by 25% every year. Time (years) 1 2 3 … t Value P A population doubles every three years. Time (years) 1 2 3 … t Value P

Exponential Growth and Decay A certain bacteria population doubles in size every 12 hours. By how much (by what factor) will it grow in 2 days?

Half-Life The half-life of a radioactive substance is the amount of time that it takes for exactly half of the original substance to remain. The half-life of an element is 5 years. Time (years) 5 10 15 … t Value P

Exponential Growth and Decay The half-life of Carbon-14 (C-14) is 5730 years. How much of a 10 mg sample will remain after 4500 years?

Exponential Growth and Decay Practice Problems: $100 is invested at 7.2% interest compounded quarterly. Determine how much the investment is worth after 5 years. The value of a new $3500 sailboat decreases 10% per year. Find its value after 10 years. A culture of yeast doubles in size every 20 minutes. The size of the culture now is 80. Find its size in 3 hours. The radioactive gas radon has a half-life of approximately 3.5 days. About how much of a 100 mg sample will remain after 1 week?

Learning Log Summary LT 12 – I can use exponential and logarithmic functions to model and solve growth and decay problems. When solving for time in a growth/decay problem… To write a growth or decay model…

Closure Homework: Pg. 486 ~ 1-11 (Odd)