1. 5.5 Logarithmic Functions Objective To Define and apply logarithms

2. Logarithmic Functions x = 2 y is an exponential equation. Its inverse (solving for y) is called a logarithmic equation. Let’s look at the parts of each type of equation: Exponential Equation x = a y exponent base number /logarithm y = log a x Logarithmic Equation if and only if

3. Example 1: Rewrite in exponential form and solve log a 64 = 2 a 2 = 64 a =  8 Example: Solve log 5 x = 3 Rewrite in exponential form: 5 3 = x x = 125 basenumberexponent

4. Example 2: Solve 7 y = 1/49 y = –2 An equation in the form y = log b xwhere b > 0 and b  1 is called a logarithmic function. Logarithmic and exponential functions are inverses of each other log b y = x, y = b x, log b b x = x b y = x, y = log b x, b log b x = x

5. Example 3. Evaluate each: a. log 8 8 4 b. 6 [log 6 (3y – 1)] log b b x = x log 8 8 4 = 4 b log b x = x 6 [log 6 (3y – 1)] = 3y – 1 Here are some special logarithm values: 1. log a 1 = 0 because a 0 = 1 2. log a a = 1 because a 1 = a 3. log a a x = x because a x = a x

6. Example 4 : Find

7. The logarithm with base 10 is called the common logarithmic (this is the one your calculator evaluates with the log key). To use a calculator to evaluate logarithms with other bases, you can change the base to 10 by using the following formula: Change of Base Formula: For all positive numbers a, b, and n, where a ≠ 1 and b ≠ 1, Example: Approximate log 4 22 ≈ 2.2295

8. Example 5. Two loud stereos are playing the same music simultaneously at 80 dB each. What is the decibel level of the combined sound? By how many decibels is the decibel level of the two stereos greater than the decibel level of one stereo?

9. The logarithm with base e is called the natural logarithmic (this is the one your calculator evaluates with the ln key). To use a calculator to evaluate logarithms with other bases, you can change the base to e by using the following formula: Change of Base Formula: For all positive numbers a, b, and n, where a ≠ 1 and b ≠ 1, Example: Approximate log 3 50 ≈ 3.56088

10. The base b logarithmic function is the inverse of the base b exponential function. Domain ofAll reals Range ofPositive reals Domain ofPositive reals Range ofAll reals The most important logarithmic function in advanced mathematics and statistics has the number e as its base. The natural logarithm of x is usually denoted ln x although sometimes it is written if and only if For example ln 5  1.6 because e 1.6 = 5

11. Example 6. Find the value of x to the nearest hundredth.

12. How do you graph a logarithmic function? Example 7: Graph f(x) = log 3 x This is the inverse of g(x) = 3 x We will need to create a table of values. (Keep in mind that logarithmic functions are inverses of exponential functions) x g(x) -2 0 1 2 1/9 1/3 1 3 9 x f(x) -2 0 1 2 1/9 1/3 1 3 9 f(x) = log 3 x g(x) = 3 x

13. Assignment P. 194 #2 – 18 (even), 35 – 49 (odd)