Intro to Logarithms Goes along with 4.4 (GREEN book) Quiz: 1/12/10 Logs Test: 1/21/10

Introduction Logarithms were originally developed to simplify complex arithmetic calculations. They were designed to transform multiplicative processes into additive ones. If at first this seems like no big deal, then try multiplying 2,234,459,912 and 3,456,234,459. (Without a calculator!) Clearly, it is a lot easier to add these two numbers.

Definition Logarithms are the "opposite" of exponentials, just as subtraction is the opposite of addition and division is the opposite of multiplication. Logs "undo" exponentials. Technically speaking, logs are the inverses of exponentials.

Lets look at their graphs y = log 10 xy = 10 x y=x

Understanding Logarithms The first, and perhaps the most important step, in understanding logarithms is to realize that they always relate back to exponential equations. You must be able to convert an exponential equation into logarithmic form and vice versa.

Exponential Logarithmic Equation Form 10 3 = 1000 log = 3 BASE Exponent (The base of the logarithm must be a positive number other than 1.) (You can’t take the log of a negative number or zero.)

log x y y b x =

Example 1: Write 5 3 = 135 in logarithmic form. Write log 3 81 = 4 in exponential form.

Try This: Complete the table. Exponential Form 2 5 = = 1/9 Logarithmic Form log = 3Log 16 4 = 1/2

Evaluate without a calculator: 1. log log Find the value of k : k = log Find the value of k : ½ = log k 9 5. Find the value of k : 3 = log 7 k

Common Logarithms Logarithms with base ______ are called common logarithms. Sometimes the base is assumed and not written. Thus, if you see a log written without a base, you assume the base is _______. The log button the calculator uses base _____. 10

Use your calculator to evaluate: 1. log log 4 3. log Solve for x: 10 x = Solve for x: 10 x = 1/1085

Natural Logarithm A natural logarithm is a logarithm with base e, denoted by ln. A natural logarithm is the inverse of an exponential function with base e.

Lets look at their graphs y = e x y = ln x y=x

Homework Green Book P. 145 Guided Practice #1-7odd P. 147 #1-12 all