More on Logarithmic Functions 9.6

Logarithmic Functions A logarithm is the inverse of an exponential function. y = bx x = by To find the inverse, switch x and y Exponential function Logarithmic function

Logarithmic Functions We write this function like this: The base of the exponent and log y = logbx The answer to a log is a power The “answer” to by means x = by

Logarithmic vs Exponential Equations power ”answer” y = logbx base Rewrite each exponential equation as a log means Rewrite each logarithmic equation as an exponential equation means

y = logbx Evaluating Logs power Question to ask: What power of b is x? ”answer” y = logbx base Find the value of each log. Remember, the answer to a log is an exponent! (because 23 = 8) (because 3-2 = 1/9) (because 251/2 = 5)

Common and Natural Logs Log of base 10 (log10x) is called the common log. It is written as log x. Log in base e (logex) is called the natural log. It is written as ln x. e is a constant like π. Find e on your calculator. What is its value? The values of common and natural logs can be found with a calculator.

Estimating Common and Natural Logs Estimate using a calculator:

More Properties of Logs Because e 1 = e Find the exact value: Because e 0 = 1 The natural log cancels out base e (they are inverses)

Solving Logarithmic Equations power ”answer” y = logbx Solve: base means Base is 10 means Base is e means Base is e

Change of Base Some logs have rational values as answers. These are the ones we can do by hand. (Similar to perfect squares – we don’t need a calculator to find their square root)

Change of Base If the value of the log is irrational, we would need to estimate in order to approximate its value. For example, log29 is irrational because there is no integer value of x such that 2x = 9. Another problem: the only values of logs we can estimate using a calculator are base 10 and base e (ln). Is it possible to estimate log29?

The change of base formula is: “Answer” to previous problem a c b c Old base New base (any base you want) Yes! The change of base formula allows us to estimate logs we can’t do by hand.

Since we can use any base in the formula, if the objective is to estimate, which bases should we choose? The natural log Log in base 10 Use either: OR

Verifying solution: 43.3219 is about 100! Estimate the log using the change of base formula: 3.170 The natural log would also work: 3.170 You try: estimate to 4 decimal places using change of base formula 3.3219 Verifying solution: 43.3219 is about 100!