Chapter 11 Sec 4 Logarithmic Functions

2 of 16 Pre-Calculus Chapter 11 Sections 4 & 5 Graph an Exponential Function If y = 2 x we see exponential growth meaning as x slowly increases y grows rapidly. The inverse of this function is x = 2 y this represent quantities that increase or decrease slowly.. In general the inverse of y = b x is x = b y. y is called the logarithmof x and is usually written as y = log b x and is read log base b of x. x = b y y is called the logarithm of x and is usually written as y = log b x and is read log base b of x

3 of 16 Pre-Calculus Chapter 11 Sections 4 & 5 Logarithm with Base b

4 of 16 Pre-Calculus Chapter 11 Sections 4 & 5 Logarithmic to Exponential Form Write each expression in exponential form. log b N = k if and only if b k = N a. log 8 1 = 0 b = 8 N = 1 k = 0 b. log = 3 b = 5 N = 125 k = 3 c. log = 2 b = 13 N = 169 k =2 b = 2 N = 1/16 k = = = = 169

5 of 16 Pre-Calculus Chapter 11 Sections 4 & 5 Exponential to Logarithmic Form Write each expression in logarithmic form. log b N = k if and only if b k = N a = 1000 b = 10 N = 1000 k = 3 b. 3 3 = 27 b = 3 N = 27 k = 3 b = 1/3 N = 9 k = - 2 b = 9 N = 3 k =1/2 log = 3 log 3 27 = 3

6 of 16 Pre-Calculus Chapter 11 Sections 4 & 5 Evaluate Logarithmic Expressions Evaluate log 2 64, remember log b N = k and b k = N so..find k a. log k = 64 2 k = 2 6 so… k = 6 Now, log 2 64 = 6 a. log k = k = 3 5 so… k = 5 Now, log = 5 = k

7 of 16 Pre-Calculus Chapter 11 Sections 4 & 5 Evaluate Logarithmic Expressions Evaluate each expression. log b N = k and b k = N a. log log = k 6 k = 6 8 so… k = 8 log = 8 b =3 k = log 3 (4x - 1) log 3 N = log 3 (4x - 1) so… N = 4x -1

8 of 16 Pre-Calculus Chapter 11 Sections 4 & 5Properties

9 of 16 Pre-Calculus Chapter 11 Sections 4 & 5Example Solve each equation X

Chapter 11 Sec 5 Common Logarithm

11 of 16 Pre-Calculus Chapter 11 Sections 4 & 5 Common Logs Common Logarithms are all logarithms that have a base of 10…log 10 x = log 3 Common Logarithms are all logarithms that have a base of 10…log 10 x = log 3 Most calculators have a key for evaluation common logarithms. LOG Example 1. Use a calculator to evaluate each expression to four decimal places. a. log 3 b. log 0.2 LOG 3 ENTER.4771 LOG 0.2 ENTER –.6990

12 of 16 Pre-Calculus Chapter 11 Sections 4 & 5Solving Solve 3 x = 11 3 x = 11 log 3 x = log 11 x log 3 = log 11 Equality property Power property Divide each side by log 3 Solve 5 x = 62 5 x = 62 log 5 x = log 62 x log 5 = log 62

13 of 16 Pre-Calculus Chapter 11 Sections 4 & 5 Change of Base Formula This allows you to write equivalent logarithmic expressions that have different bases. For example change base 3 into base 10 This allows you to write equivalent logarithmic expressions that have different bases. For example change base 3 into base 10

14 of 16 Pre-Calculus Chapter 11 Sections 4 & 5 Change of Base Express log 4 25 in terms of common logarithms. Then approximate its value.

15 of 16 Pre-Calculus Chapter 11 Sections 4 & 5Antilogarithm Sometime the logarithm of x is know to have a value of a, but x is not known. Sometime the logarithm of x is know to have a value of a, but x is not known. Then x is called the antilogarithm of a, written as antilog a. Then x is called the antilogarithm of a, written as antilog a. So, if log x = a, then x = antilog a. So, if log x = a, then x = antilog a. Remember that the inverse (or antilog) of a logarithmic function is an exponential function. Remember that the inverse (or antilog) of a logarithmic function is an exponential function. ie log x = 2.7 → x = antilog 2.7 or x =501.2

16 of 16 Pre-Calculus Chapter 11 Sections 4 & 5 Daily Assignment Chapter 11 Sections 4 & 5 Text Book Pgs 723 – 724 #21 – 51 Odd; Pgs 730 – 731 #19 – 45 Odd;