Essential Question: What is the relationship between a logarithm and an exponent?

 You’ve ran across a multitude of inverses in mathematics so far... ◦ Additive Inverses: 3 & -3 ◦ Multiplicative Inverses: 2 & ½ ◦ Inverse of powers: x 4 & or x ¼ ◦ But what do you do when the exponent is unknown? For example, how would you solve 3 x = 28, other than guess & check? ◦ Welcome to logs…

 Logs ◦ There are three types of commonly used logs  Common logarithms (base 10)  Natural logarithms (base e)  Binary logarithms (base 2) ◦ We’re only going to concentrate on the first two types of logarithms, the 3 rd is used primarily in computer science. ◦ Want to take a guess as to why I used the words “base” above?

 The logarithm to the base b of a positive number y is defined as follows: ◦ If y = b x, then log b y = x ◦ All logs can be thought of as a way to solve for an unknown exponent  log base answer = exponent  log 10 2 x = x 2 x =

 Example: Write “25 = 5 2 ” in logarithmic form ◦ Remember: log base answer = exponent ◦ log 5 25 = 2  Your turn: Write the following exponential equations in logarithmic form. ◦ 729 = 3 6 ◦ ( 1 / 2 ) 3 = 1 / 8 ◦ 10 0 = 1 log = 6 log 1 / 2 1 / 8 = 3 log 10 1 = 0

 Example: Write “log 8 16 = x” in exponential form ◦ Remember: log base answer = exponent ◦ 8 x = 16  Your turn: Write the following exponential equations in logarithmic form. ◦ log 64 1 / 32 = x ◦ log 9 27 = x ◦ log = x 64 x = 1 / 32 9 x = x = 100

 Assignment ◦ Page 450 ◦ Problems 7 – 25 & 53 – 61 (odd problems)  For questions 15 – 25, pretend they “= x”  We’ll deal with how to solve them Tuesday

Essential Question: What is the relationship between a logarithm and an exponent?

 Common logarithms ◦ Scientific/graphing calculators have the common and natural logarithmic tables built in. ◦ On our calculators, the “log” button is next to the carat (^) key. ◦ To find log 10 29, simply type “log 29”, and you will be returned the answer  That means, = 29 ◦ Though the calculator will give you logs to a bunch of places, round your answers to 4 decimal places

 Solving Logarithmic Equations (w/o calculator) ◦ log 2 16 = x  Can be rewritten as 2 x = 16.  Because 2 4 = 16, x = 4 ◦ log 5 (-25) = x  Rewritten as 5 x = -25, which isn’t possible.  Undefined ◦ log 5 x = 3  Can be rewritten as 5 3 = x, so x = 125

 Change-of-Base Formula ◦ ◦ Find log 8 9 

 Example: Solve for x log 8 16 = x  Your turn: Solve for x ◦ log 64 1 / 32 = x ◦ log 9 27 = x ◦ log = x

 Assignment ◦ Page 450 ◦ Solve problems 41 – 47, odds ◦ Solve problems 15 – 25, odds ◦ Page 464 ◦ Solve problems 25 – 31, odds