7.2 Properties of Logarithms (Review) Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2004 Golden Gate Bridge San Francisco, CA

Many real-life phenomena can be modeled by an exponential function with base, where. e can be approximated by:

Consider This is a one-to-one function, therefore it has an inverse. The inverse is called a logarithm function. Example: Two raised to what power is 16? The most commonly used bases for logs are 10: and e : is called the natural log function. is called the common log function.

is called the natural log function. is called the common log function. In calculus we will primarily use natural logs. We will have ways to find derivatives and integrals of natural logs Common logs can be changed to natural logs.

Properties of Logarithms Since logs and exponentiation are inverse functions, they “un-do” each other. Product rule: Quotient rule: Power rule: Change of base formula:

Example 1: Find the exact value without using a calculator: Using properties of logarithims.

Example 2: Expand the logarithm in terms of sums, differences, and multiples of simpler logarithms:

Example 3: Solve for x :

Example 4: Solve for x :

Example 5: $1000 is invested at 5.25 % interest compounded annually. How long will it take to reach $2500? We use logs when we have an unknown exponent years In real life you would have to wait 18 years.