5.1 Exponential Functions

Our presentation today will consists of two sections. Section 1: Exploration of exponential functions and their graphs. Section 2: Discussion of the equality property and its applications.

First, let’s take a look at an exponential function y 1 2 4 -1 1/2 -2 1/4

First let’s change the base b to positive values What conclusion can we draw ?

Next, observe what happens when b assumes a value such that 0<b<1. Can you explain why this happens ?

will happen if ‘b’ is negative ? What do you think will happen if ‘b’ is negative ?

Don’t forget our definition ! Any equation of the form: Can you explain why ‘b’ is restricted from assuming negative values ?

To see what impact ‘a’ has on our graph we will fix the value of ‘b’ at 3. What does a larger value of ‘a’ accomplish ?

Let’s see if you are correct ! Shall we speculate as to what happens when ‘a’ assumes negative values ? Let’s see if you are correct !

Our general exponential form is “b” is the base of the function and changes here will result in: When b>1, a steep increase in the value of ‘y’ as ‘x’ increases. When 0<b<1, a steep decrease in the value of ‘y’ as ‘x’ increases.

We also discovered that changes in “a” would change the y-intercept on its corresponding graph. Now let’s turn our attention to a useful property of exponential functions.

The Equality Property of Exponential Functions Section 2 The Equality Property of Exponential Functions

We know that in exponential functions the exponent is a variable. When we wish to solve for that variable we have two approaches we can take. One approach is to use a logarithm. We will learn about these in a later lesson. The second is to make use of the Equality Property for Exponential Functions.

The Equality Property for Exponential Functions This property gives us a technique to solve equations involving exponential functions. Let’s look at some examples. Basically, this states that if the bases are the same, then we can simply set the exponents equal. This property is quite useful when we are trying to solve equations involving exponential functions. Let’s try a few examples to see how it works.

Here is another example for you to try: (Since the bases are the same we simply set the exponents equal.) Here is another example for you to try: Example 1a:

The next problem is what to do when the bases are not the same. Does anyone have an idea how we might approach this?

Here for example, we know that Our strategy here is to rewrite the bases so that they are both the same. Here for example, we know that

Example 2: (Let’s solve it now) (our bases are now the same so simply set the exponents equal) Let’s try another one of these.

Example 3 Remember a negative exponent is simply another way of writing a fraction The bases are now the same so set the exponents equal.

By now you can see that the equality property is actually quite useful in solving these problems. Here are a few more examples for you to try.

SIMPLE AND COMPOUND INTEREST Since this section involves what can happen to your money, it should be of INTEREST to you!

I = PRT IMPLE INTEREST FORMULA Annual interest rate Interest paid Time (in years) Principal (Amount of money invested or borrowed)

If you invested $200.00 in an account that paid simple interest, find how long you’d need to leave it in at 4% interest to make $10.00. enter in formula as a decimal I = PRT 10 = (200)(0.04)T 1.25 yrs = T Typically interest is NOT simple interest but is paid semi-annually (twice a year), quarterly (4 times per year), monthly (12 times per year), or even daily (365 times per year).

COMPOUND INTEREST FORMULA annual interest rate (as a decimal) Principal (amount at start) time (in years) amount at the end number of times per year that interest in compounded

Find the amount that results from $500 invested at 8% compounded quarterly after a period of 2 years. 4 (2) .08 500 4 Effective rate of interest is the equivalent annual simple rate of interest that would yield the same amount as that made compounding. This is found by finding the interest made when compounded and subbing that in the simple interest formula and solving for rate. Find the effective rate of interest for the problem above. The interest made was $85.83. Use the simple interest formula and solve for r to get the effective rate of interest. I = Prt 85.83=(500)r(2) r = .08583 = 8.583%

5.2 Natural Exponential Functions Continuous Compound Interest A = the accumulated amount after t years P = Principal r = Nominal interest rate per year t = Number of years

5.2 Natural Exponential Functions The Natural Base e Many applied exponential functions involves the irrational number, 2.71828182845904… symbolized by the letter e. The number e is called the natural base. The function f(x) = ex is called the natural exponential function.

5.2 Find the accumulated amount of money after 25 years if $7500 is invested at 12% per year compounded continuously.

5.2 An investment of $10,000 increased to $28,576.51 in 15 years. If interest was compounded continuously, find the interest rate. A = Pert

5.2 Model of Population Growth If b is the annual birth rate, d is the annual death rate, t is the time (in years), P0 is the initial population at t = 0, and P is the current population, then P = P0ekt where k = b - d is the annual growth rate, the difference between the annual birth rate and death rate.

5.2 The population of a city in 1970 was 153,800. Assuming the population increases continuously at a rate of 5% per year, predict the population of the city in 2010.

5.2 If f(x) = x2(-2e-2x) + 2xe-2x, find the zeros.

5.2 Assignment pp. 345-8 (1-25 odd) Transparencies: 19, 21, 23, 25

5.3 Introduction To Logarithms

Logarithms were originally developed to simplify complex arithmetic calculations. They were designed to transform multiplicative processes into additive ones.

Definition of Logarithm Suppose b>0 and b≠1, there is a number ‘p’ such that:

So let’s get a lot of practice with this ! You must be able to convert an exponential equation into logarithmic form and vice versa. So let’s get a lot of practice with this !

We read this as: ”the log base 2 of 8 is equal to 3”. Example 1: Solution: We read this as: ”the log base 2 of 8 is equal to 3”.

Read as: “the log base 4 of 16 is equal to 2”. Example 1a: Solution: Read as: “the log base 4 of 16 is equal to 2”.

Example 1b: Solution:

Okay, so now it’s time for you to try some on your own.

Solution:

Solution:

Solution:

This is simply the reverse of It is also very important to be able to start with a logarithmic expression and change this into exponential form. This is simply the reverse of what we just did.

Example 1: Solution:

Example 2: Solution:

Okay, now you try these next three.

Solution:

Solution:

Solution:

Let’s rewrite the problem in exponential form. Solution: Let’s rewrite the problem in exponential form. We’re finished !

Rewrite the problem in exponential form. Solution: Rewrite the problem in exponential form.

Solution: Example 3 Try setting this up like this: Now rewrite in exponential form.

Solution: Example 4 First, we write the problem with a variable. Now take it out of the logarithmic form and write it in exponential form.

Solution: Example 5 First, we write the problem with a variable. Now take it out of the exponential form and write it in logarithmic form.

Basically, with logarithmic functions, Finally, we want to take a look at the Property of Equality for Logarithmic Functions. Basically, with logarithmic functions, if the bases match on both sides of the equal sign , then simply set the arguments equal.

Since the bases are both ‘3’ we simply set the arguments equal. Example 1 Solution: Since the bases are both ‘3’ we simply set the arguments equal.

Solution: Example 2 Factor Since the bases are both ‘8’ we simply set the arguments equal. Factor continued on the next page

Solution: Example 2 continued It appears that we have 2 solutions here. If we take a closer look at the definition of a logarithm however, we will see that not only must we use positive bases, but also we see that the arguments must be positive as well. Therefore -2 is not a solution. Let’s end this lesson by taking a closer look at this.

Can anyone give us an explanation ? Our final concern then is to determine why logarithms like the one below are undefined. Can anyone give us an explanation ?

We’ll then see why a negative value is not permitted. One easy explanation is to simply rewrite this logarithm in exponential form. We’ll then see why a negative value is not permitted. First, we write the problem with a variable. Now take it out of the logarithmic form and write it in exponential form. What power of 2 would gives us -8 ? Hence expressions of this type are undefined.

5.3 Answers 6. t = 1/4loga10/3 12. a.) 10-8 = x; b.) 10y-2 = x c.) e1/2 =x d.) e7+x = z e.) e1.2 = t-5 18. -3/2 48.a) 78.8 24. 21 b) 92,900 30. 5 c) 0.00614 36. graph d) 40.4 e) 2.59 f) 0.00674

5.4 Properties of logarithms Objectives Use the product rule. Use the quotient rule. Use the power rule. Expand logarithmic expressions. Condense logarithmic expressions.

Logarithms of Products The Product Rule For any positive numbers M and N and any logarithmic base a, loga MN = loga M + loga N. (The logarithm of a product is the sum of the logarithms of the factors.)

Example Condense to a single logarithm: Expand to a sum of logarithms

Logarithms of Powers The Power Rule For any positive number M, any logarithmic base a, and any real number p, (The logarithm of a power of M is the exponent times the logarithm of M.)

Examples Expand to a product. Expand to a product.

Examples Condense the product to a single logarithm using exponents.

Logarithms of Quotients The Quotient Rule For any positive numbers M and N, and any logarithmic base a, . (The logarithm of a quotient is the logarithm of the numerator minus the logarithm of the denominator.)

Examples Expand to a difference of logarithms. Condense to a single logarithm.

Expand a logarithmic expression Use properties of logarithms to change one logarithm into a sum or difference of others Example

Expand in terms of sums and differences of logarithms.

Example Condense to a single logarithm.

5.5 Exponential & Logarithmic Equations Exponential Equations with Like Bases Exponential Equations with Different Bases Logarithmic Equations Change of Base Formulas

5.5 Exponential and Logarithmic Equations Objective: We will consider various types of exponential and logarithmic equations and their applications.

Change of Base Formula The 2 bases we are most able to calculate logarithms for are base 10 and base e. These are the only bases that our calculators have buttons for. For ease of computing a logarithm, we may want to switch from one base to another. The new base, a, can be any integer>1, but we often let a=10 or a=e.

Compute What is the log, base 5, of 29? Does this answer make sense? What power would you raise 5 to, to get 29? A little more than 2! (5 squared is 25, so we would expect the answer to be slightly more than 2.)

Rewrite as a quotient of natural logarithms

Example Find log6 8 using common logarithms. Solution: First, we let a = 10, b = 6, and M = 8. Then we substitute into the change-of-base formula:

Example We can also use base e for a conversion. Find log6 8 using natural logarithms. Solution: Substituting e for a, 6 for b and 8 for M, we have

Exponential Equations with Like Bases In an Exponential Equation, the variable is in the exponent. There may be one exponential term or more than one, like… If you can isolate terms so that the equation can be written as two expressions with the same base, as in the equations above, then the solution is simple. or

Exponential Equations with Like Bases Example #1 - One exponential expression. 1. Isolate the exponential expression and rewrite the constant in terms of the same base. 2. Set the exponents equal to each other (drop the bases) and solve the resulting equation.

Exponential Equations with Like Bases Example #2 - Two exponential expressions. 1. Isolate the exponential expressions on either side of the =. We then rewrite the 2nd expression in terms of the same base as the first. 2. Set the exponents equal to each other (drop the bases) and solve the resulting equation.

Exponential Equations with Different Bases The Exponential Equations below contain exponential expressions whose bases cannot be rewritten as the same rational number. The solutions are irrational numbers, we will need to use a log function to evaluate them. or

Exponential Equations with Different Bases Example #1 - One exponential expression. 1. Isolate the exponential expression. 2. Take the log (log or ln) of both sides of the equation. 3. Use the log rule that lets you rewrite the exponent as a multiplier.

Exponential Equations with Different Bases Example #1 - One exponential expression. 4. Isolate the variable.

Exponential Equations with Different Bases Example #2 - Two exponential expressions. 1. The exponential expressions are already isolated. 2. Take the log (log or ln) of both sides of the equation. 3. Use the log rule that lets you rewrite the exponent as a multiplier on each side..

Exponential Equations with Different Bases Example #2 - Two exponential expressions. 4. To isolate the variable, we need to combine the ‘x’ terms, then factor out the ‘x’ and divide.

Logarithmic Equations In a Logarithmic Equation, the variable can be inside the log function or inside the base of the log. There may be one log term or more than one. For example …

Logarithmic Equations Example 1 - Variable inside the log function. 1. Isolate the log expression. 2. Rewrite the log equation as an exponential equation and solve for ‘x’.

Logarithmic Equations Example 3 - Variable inside the base of the log. 1. Rewrite the log equation as an exponential equation. 2. Solve the exponential equation.

Ch. 5 Review Answers