if and only if Definition ylog base a

if and only if Definition y=log base a of x

Important Idea Logarithmic Form Exponential Form

Important Idea The logarithmic function is the inverse of the exponential function

Example Write the following logarithmic function in exponential form:

Important Idea In your book and on the calculator, is the same as. If no base is stated, it is understood that the base is 10.

Try This Without using your calculator, find each value: 5 1 1/3 undefined

Example Solve each equation by using an equivalent statement:

Definition A second type of logarithm exists, called the natural logarithm and written ln x, that uses the number e as a base instead of the number 10. The natural logarithm is very useful in science and engineering.

Important Idea Like , the number e is a very important number in mathematics.

Important Idea The natural logarithm is a logarithm with the base e is a short way of writing:

Definition If and only if

Try This Use a calculator to find the following value to the nearest ten-thousandth: 1. 1394

Try This Solve each equation by using an equivalent statement: x=7. 389 x=2. 079

Example Using your calculator, graph the following: Where does the graph cross the x-axis?

Example Using your calculator, graph the following: Can ln x ever be 0 or negative?

Example Using your calculator, graph the following: What is the domain and range of ln x?

Example Using your calculator, graph the following: How fast does ln x grow? Find the ln 1, 000.

Try This Using your calculator, graph: Describe the differences. How does the domain and range change?

Try This Solve for x: 1. 151 -. 077 531434 -2 , -1

Try This Solve for x: 2. 944. 564 6

Important Idea The definitions of common and natural logarithms differ only in their bases, therefore, they share the same properties and laws.

Important Idea Properties of Common Logarithms: • log x defined only for x>0 • log 1=0 & log 10=1 • • for x >0

Important Idea Properties of Natural Logarithms: • ln x defined only for x>0 • ln 1=0 & ln e=1 • • for x >0

Important Idea

MUST REMEMBER Product Law: ln(ab)=ln a + ln b Quotient Law: Power Law: ln an=n ln a

Same Rules for any base Product Law: Quotient Law: Power Law:

Express In terms of log A, log B, and Log C

Use a combination of logarithmic properties and laws to rewrite the given expression: