By Dr. Julia Arnold

Concept 1 The Exponential Function

The exponential function f with base a is denoted by f(x) = a x where a > 0 a 1, and x is any real number. To graph a specific exponential function we will use a table of values. x 2 x /2 -2 1/4 This is the graph of 2 x

Pi or is what is called a transcendental number ( which means it is not the root of some number). e x is called the natural exponential function. is another such number. On the graphing calculator, you can find e by pushing the yellow 2nd function button and the Ln key. On the display you will see e^( Type 1 and close parenthesis. Thus e^( 1). Press Enter and you will see which represents an approximation of e.

Concept 2 The exponential function is the inverse of the logarithm function

Recall that Inverse Functions reverse the ordered pairs which belong to functions. i.e. (x,y) becomes (y,x) The log function is the inverse of the exponential function. If the exponential is 2 x, then its inverse is log 2 (x) (read log x base 2 ). If the exponential is 3 x... its inverse is log 3 (x) Read log x base 3 If the exponential is 10 x... its inverse is log x Read log x Base 10 is considered the common base and thus log x is the common log and as such the base is omitted.

If the exponential is 2 x =y, then its inverse is log 2 (x)=y (read log y base 2 =‘s x ). If the exponential is 3 x... its inverse is log 3 (x) If the exponential is 10 x... its inverse is log x Base 10 is considered the common base and thus log y is the common log and as such the base is omitted. If the exponential is e x...its inverse is ln x Read l n x Base e is considered the natural base and thus ln x is the natural log and is written ln to distinguish it from log.

Concept 3 How to graph the logarithmic function.

Let’s look at the two graphs of the exponential and the logarithmic function: f(x)=2 x f(x)=log 2 (x) Goes through (0,1) which means 1 = 2 0 Goes through (1,0) which means 0 = log 2 (1) The domain is all real numbers. The domain is all positive real numbers. The range is all positive real numbers. The range is all real numbers.

Remember this slide? We used a table of values to graph the exponential y = 2 x. x 2 x /2 -2 1/4 f(x)=2 x Since we know that the y = log 2 (x) is the inverse of the function above, we can just switch the ordered pairs in the table above and create the log graph for base 2.

x 2 x /2 -2 1/4 f(x)=2 x y = log 2 (x) is the inverse of the function y = 2 x. To create the graph we can just switch the ordered pairs in the table left and create the log graph for base 2. xlog 2 (x) /2 -1 1/4 -2 Switch This is the easy way to do a log graph.

Concept 4 Changing from exponential form to logarithmic form.

First task is to be able to go from exponential form to logarithmic form. x = a y becomes y = log a (x) Both of these are referred to as bases Y is the exponent on the left. Logs are = to the exponent on the right. Thus, 2 5 = 32 becomes log 2 (32)=5 Read: log 32 base 2 =‘s 5

Thus, 2 5 = 32 becomes log 2 (32) = = 1000 becomes log (1000) = 3 e 1 = e becomes ln (e) = = 81 becomes log 3 (81) = = 1/8 becomes log 2 (1/8) = = 1/25 becomes log 5 (1/25) = = 1 becomes log 2 (1) = 0

Concept 5 Four log properties.

There are a few truths about logs which we will call properties: 1. log a (1) = 0 for any a > 0 and not equal to 1 because a 0 =1 (exponential form of log form) 2. log a (a) = 1 for any a > 0 and not equal to 1 because a 1 =a (exponential form of log form) 3. log a (a x ) = x for any a > 0 and not equal to 1 because a x =a x (exponential form of log form) 4. If log a x = log a y, then x = y.

Practice Problems 1. Solve for x: log 3 x = log 3 4 X = 3 4 X = 4 Click on the green arrow of the correct answer above.

No, x = 3 4 is not correct. Use the 4th property: 4. If log a x = log a y, then x = y. log 3 x = log 3 4, then x = 4 Go back.

Way to go! Using the 4th property: 4. If log a x = log a y, then x = y you concluded correctly that x = 4 for log 3 x = log 3 4.

Practice Problems 2. Solve for x: log 2 1/8 = x X = 3 X = -3 Click on the green arrow of the correct answer above.

No, x = 3 is not correct. Use the 3rd property: 3. log a (a x ) = x log 2 1/8 = log = log 2 (2 3 ) -1 = log then x = -3 since the 2’s make a match. Go back.

Way to go! Using the 3rd property: 3. log a (a x ) = x log 2 1/8 = log = log 2 (2 3 ) -1 = log then x = -3 since the 2’s make a match.

Practice Problems 3. Evaluate: ln 1 + log 10 - log 2 (2 4 ) Click on the green arrow of the correct answer above

No, -2 is not correct. Using properties 1,2 and 3: ln 1 = 0 log 10 = 1 log = -4 which totals to -3 Go back.

Way to go! Using properties 1,2 and 3: ln 1 = 0 log 10 = 1 log = -4 which totals to -3

Concept 6 The three expansion properties of logs.

The 3 expansion properties of logs 1. log a (uv) = log a u + log a v Proof: Set log a u =x and log a v =y then change to exponential form. a x = u and a y = v. a x+y =a x a y = uv so, write a x+y = uv in log form log a (uv) = x + y but that’s log a u =x and log a v =y, so write log a (uv) = log a u + log a v which is the result we were looking for. Do you see how this property relates to the exponential property?

The 3 expansion properties of logs 1. log a (uv) = log a u + log a v 2. 3.

Example 1 Expand to single log expressions: Applying property 1 Log 10 = 1 from the 2nd property which we had earlier.

Example 2 Expand to single log expressions: Applying property 2 Applying property 1 and from before ln e = 1

Example 3 Expand to single log expressions: First change the radical to an exponent. Next, apply property 3

Next, apply property 2 for quotients

Next, apply property 3to the exponentials. Now they are single logs.

Your turn: Expand to single logs: The first step is to use property 1 The first step is to use property 3

No, incorrect, return to the previous slideprevious slide.

The first step is to use property 1 which will expand to: Now we use property 3 This is the final answer.

No, this is not the final answer. Return to the previous slide andprevious slide click on the correct answer.

Yes, we now use property 3 to expand further to: This is the final answer. This is still not the final answer.

Nope, we are not done yet. Return

Whenever the base of the log matches the number you are taking the log of, the answer is the exponent on the number which is 1 in this case.

is the property. Or from the beginning of the problem you could have said:

We can use the same 3 expansion properties of logs to take an expanded log and condense it back to a single log expression. 1. log a u + log a v =log a (uv) 2. 3.

Condense to a single log: Always begin by reversing property 3 Next use property 2 which is a single log

Condense to a single log: reversing property 3 Next use property 2 which is a single log

Concept 7 Finding logs on your calculator for any base number.

On your graphing calculator or scientific calculator, you may find the value of the log (of a number) to the base 10 or the ln(of a number) to the base e by simply pressing the appropriate button. What if you want to find the value of a log to a different base? How can we find for example, the log 2 5

Set log 2 5 = x This shows us how we can create the change of base formula: Changes the base to c. Change to exponential form 2 x = 5 Take the log (base 10 ) of both sides. log 2 x = log 5 x log 2 = log 5 using the 3rd expansion property thus x =

The change of base formula: We are given base a, and we change to base c.

Example 1 Find the following value using a calculator: Since the calculator is built to find base 10 or base e, choose either one and use the change of base formula.

Some problems can be done without a calculator, but not all. 1. Find using the property 2. Find using the same property

Some problems can be done without a calculator, but not all. 1. Find Click on the correct slide to advance to the next slide. Click on the wrong side and you will remain here.

Right! Find ln 10

Right! while as found using a calculator. Which of the following is false concerning None are false

Because of the log properties all of the statements below are true Uses the change of base formula

Concept 8 Solving exponential equations with logs

Solve Take the ln of both sides. Property: lne x = x Solve Isolate the exponential expression. Take the log base 10 of both sides.

Solve Isolate the exponential expression. Property: lne x = x Take the ln of both sides

Solve Isolate the exponential expression. Take the log base 10 of both sides.

Concept 9 Solving logarithm equations with exponents

Solve Remember property 4. If log a x = log a y, then x = y. That property applies in this problem, thus x - 1 = 3 x = 4 Solve Change to exponential form

Solve Apply the expansion property for exponents. Isolate the ln expression. Change to the exponential form Solve for x

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