Inverse Functions ; Exponential and Logarithmic Functions (Chapter4) University of Palestine IT-College College Algebra Inverse Functions ; Exponential and Logarithmic Functions (Chapter4) L:18 1

Definition and Graph of the Natural Exponential Function

The Natural Base e An irrational number, symbolized by the letter e, appears as the base in many applied exponential functions. This irrational number is approximately equal to 2.72. More accurately, The number e is called the natural base. The function f (x) = ex is called the natural exponential function. -1 f (x) = ex f (x) = 2x f (x) = 3x (0, 1) (1, 2) 1 2 3 4 (1, e) (1, 3)

  Solving Exponential Equations,  where x is in the exponent, BUT the bases DO NOT MATCH. Step 1: Isolate the exponential expression.   Get your exponential expression on one side everything outside of the exponential expression on the other side of your equation. Step 2: Take the natural log of both sides. The inverse operation of an exponential expression is a log.  Make sure that you do the same thing to both sides of your equation to keep them equal to each other. Step 3: Use the properties of logs to pull the x out of the exponent. Step 4: Solve for x. Now that the variable is out of the exponent, solve for the variable using inverse operations to complete the problem.

Example 1: Solve the exponential equation Round your answer to two decimal places. Step 1: Isolate the exponential expression. This is already done for us in this problem. Step 2: Take the natural log of both sides.   Step 3: Use the properties of logs to pull the x out of the exponent.

Step 4: Solve for x.

Example 2: Solve the exponential equation Round your answer to two decimal places. Step 1: Isolate the exponential expression. Step 2: Take the natural log of both sides.   Step 3: Use the properties of logs to pull the x out of the exponent.

Step 4: Solve for x.

Example 3: Solve the exponential equation Round your answer to two decimal places. Step 1: Isolate the exponential expression.              Step 2: Take the natural log of both sides. Step 3: Use the properties of logs to pull the x out of the exponent.

Step 4: Solve for x.

Exponential Equations Solve

Another Example Solve 3x + 1 = 27x  3

sections 4.5,4.6,4.7 & Equations Logarithmic Functions Objectives: After completing this tutorial, you should be able to: Know the definition of a logarithmic function.  Write a log function as an exponential function and vice versa.  Graph a log function.  Evaluate a log.  Be familiar with and use properties of logarithms in various situations. Solve logarithmic equations.  

Definition of Log Function For all real numbers y, and all positive numbers a (a > 0) and x, where a  1: Meaning of logax A logarithm is an exponent; logax is the exponent to which the base a must be raised to obtain x. (Note: Logarithms can be found for positive numbers only) A LOG IS ANOTHER WAY TO WRITE AN EXPONENT. 

Location of Base and Exponent in Exponential and Logarithmic Forms Logarithmic form: y = logb x Exponential Form: by = x. Exponent Exponent Base Base

Example : Express the logarithmic equation exponentially We want to use the definition that is above:                 if and only if        .                           

Examples Write each equation in its equivalent exponential form. a. 2 = log5 x b. 3 = logb 64 c. log3 7 = y Solution With the fact that y = logb x means by = x, a. 2 = log5 x means 52 = x. Logarithms are exponents. b. 3 = logb 64 means b3 = 64. Logarithms are exponents. c. log3 7 = y or y = log3 7 means 3y = 7.

Evaluating Logs Step 1: Set the log equal to x. Step 2:  Use the definition of logs shown above to write the equation in exponential form.  Step 3: Find x. Whenever you are finding a log, keep in mind that logs are another way to write exponents.  You can always use the definition to help you evaluate. 

Example : Evaluate the expression without using a calculator. Evaluating Logs Example :  Evaluate the expression without using a calculator.            

Text Example Evaluate a. log2 16 b. log3 9 c. log25 5 Solution log25 5 = 1/2 because 251/2 = 5. 25 to what power is 5? c. log25 5 log3 9 = 2 because 32 = 9. 3 to what power is 9? b. log3 9 log2 16 = 4 because 24 = 16. 2 to what power is 16? a. log2 16 Logarithmic Expression Evaluated Question Needed for Evaluation Logarithmic Expression

Graphing Log Functions

Graphing Log Functions

Characteristics of the Graph of f(x) = logax The points (1, 0), and (a, 1) are on the graph. If a > 1, then f is an increasing function; if 0 < a < 1, then f is a decreasing function. The y-axis is a vertical asymptote. The domain is (0, ), and the range is (, ).

Example 1 4 2 1/16 1 y x Graph Write in exponential form as Now find some ordered pairs. 1 4 2 1/16 1 y x

Example Graph Write in exponential form as Now find some ordered pairs. 1 0.2 1 5 y x

Translated Logarithmic Functions Graph the function. The vertical asymptote is x = 1. To find some ordered pairs, use the equivalent exponent form.

Translated Logarithmic Functions continued Graph To find some ordered pairs, use the equivalent exponent form.

Properties of Logarithms, For x > 0, y > 0, a > 0, a  1, and any real number r: The logarithm of a number raised to a power is equal to the exponent multiplied by the logarithm of the number. Power Property The logarithm of the quotient of two numbers is equal to the difference between the logarithms of the numbers. Quotient Property The logarithm of a product of two numbers is equal to the sum of the logarithms of the numbers Product Property Description Property

Using the Properties of Logarithms Rewrite each expression. Assume all variables represent positive real numbers with a  1 and b  1. a) b) c)

Using the Properties of Logarithms Write each expression as a single logarithm with coefficient 1. Assume all variables represent positive real numbers with a  1 and b  1. a) b)

Using the Properties of Logarithms Expand as much as possible. Evaluate without a calculator where possible

Inverse Properties of Logarithms Inverse Property I For a > 0, a  1: By the results of this theorem:

Inverse Properties of Logarithms Inverse Property II For b > 0, b  1: By the results of this theorem: b logb x = x           , 

Basic Logarithmic Properties Involving One Logb b = 1 because 1 is the exponent to which b must be raised to obtain b. (b1 = b). Logb 1 = 0 because 0 is the exponent to which b must be raised to obtain 1. (b0 = 1).

Properties of Common Logarithms General Properties Common Logarithms 1. logb 1 = 0 1. log 1 = 0 2. logb b = 1 2. log 10 = 1 3. logb bx = x 3. log 10x = x 4. b logb x = x 4. 10 log x = x Examples of Logarithmic Properties log b b = 1 log b 1 = 0 log 4 4 = 1 log 8 1 = 0 3 log 3 6 = 6 log 5 5 3 = 3 2 log 2 7 = 7

Properties of Natural Logarithms Logarithms with a base of e are referred to a natural logarithms. So if f(x) = ex , then f(x) = loge x = lnx Recall, e = 2.71828 Properties of Natural Logarithms General Natural Properties Logarithms 1. logb 1 = 0 1. ln 1 = 0 2. logb b = 1 2. ln e = 1 3. logb bx = 0 3. ln ex = x 4. b logb x = x 4. e ln x = x Examples log e e = 1 log e 1 = 0 e log e 6 = 6 log e e 3 = 3

Change-of-Base Theorem For any positive real numbers x, a, and b, where a  1 and b  1: logax =lnx/ lna

Examples Use the change-of-base theorem to find an approximation to four decimal places for each logarithm. a) log512 b) log2.4

Solving Exponential or Logarithmic Equations

Solving Logarithmic Equations Solve each equation. a) b)

Example Solve 8x = 15 The solution set is {1.3023}.

Example Solve continued

Example Solve

Example Solve The only valid solution is x = 4.

Example Solve

The only valid solution is x = 2. Example Solve continued The only valid solution is x = 2.

Notes, review

Properties of Logarithms

Write the following expression as the sum and/or difference of logarithms. Express all powers as factors.

Write the following expression as a single logarithm.

Most calculators only evaluate logarithmic functions with base 10 or base e. To evaluate logs with other bases, we use the change of base formula.

Practices

No Solution

End of the Lecture Let Learning Continue