Solving Exponential Equations… How do we solve these equations for x? a. b. c.

Section 4.4 Logarithmic Functions Definition Special Logarithms : Base 10 and Base e Change of Base formula Inverse Function Graphing the log function More log properties Solving Equations

I. Definition of Logarithmic Functions A logarithmic function with base a is denoted: where a > 0 and a ≠ 1 and is defined by if and only if The exponential form of is The logarithmic form of is What power of a gives you the value ? The answer:

Practice : Rewrite the expression Write the exponential expression into an equivalent logarithmic form. Ask: What power of 4 produces the number 16 ? Answer: The logarithm of 16, base 4 Ask: What power of ______ produces the number ____? Answer: The logarithm of ____, base _____.

Practice : Rewrite the expression Write the logarithmic expression into an equivalent exponential form. Find the exact value (without a calculator). 3 raised to what power ___ equals 9 ? 2 raised to what power ___ equals 1 ? ½ raised to what power ___ equals 16 ?

2. Special Logarithms - Common Log If a base is not indicated, it is understood to be 10. Example: is equivalent to

2. Special Logarithms – Natural Log natural logarithm function is expressed using the special symbol ln (logarithmus naturalis), instead of the log symbol Example: is equivalent to

Practice: Solve exponential equations and evaluate log expressions Review: #58 p. 284, #59, 60, 62, 63,66 Solve exponential expression p. 297 #33-37,39,41-43 Determine the value of the logarithmic expression.

3. Change-of-Base Formula Your calculator can compute only base 10 and base e. Use this formula to get an approximation for a logarithm to base neither 10 nor e. Example. Find an approximation for

4. Inverse Properties of Logarithmic and Exponential Functions The Logarithmic and Exponential Functions are inverses of each other. Inverse Property of Example of the relationship: Let

5. Graphing Logarithmic Functions Exponential functions and log functions are inverse functions of each other. Domain: Range: Key Points: Asymptotes:

5. A) Domain of a logarithmic function Determine the domain for these functions. Practice: Worksheet 4.3-4.4 and p. 297 #46-56 (even), 12

6) A Special Property of Logarithms If then M = N and if M = N then NOTE: We typically use base e (natural log) when applying the log to both sides of the equation.

7. Solving Logarithmic Equations. Logarithmic Equations(always check answer against domain of problem) Equal base on each side: 2. Constant on one side, logarithm on other: Use the property: then M = N Use the definition: CHECK IT! Logarithms are only defined for positive real numbers! Exclude solutions that produce logarithm of a number

8. Solving Exponential Equations. Equal base on each side: Constant on one side, exponential on other. Two ways to solve. a) Method 1 : b) Method 2: 3. Quadratic in form: Use the property: Use the definition: Take ln (natural log) of each side. We will look at this method after Section 4.5

9. Application The formula models the population of Florida, A, in millions, x years after 2000. Suppose the population is 16.3 million in 2001. Determine the population of Florida in the year 2010. When will the population reach 25.2 million ?