1. PRECALCULUS I LOGARITHMIC FUNCTIONS Dr. Claude S. Moore Danville Community College

2. DEFINITION The logarithmic function is f(x) = log a x where y = log a x iff x = a y with any real number y, x > 0, and 0 < a  1.

3. SPECIAL PROPERTIES 1. log a 1 = 0 because a 0 = 1. 2. log a a = 1 because a 1 = a. 3. log a a x = x because a x = a x 4. If log a x = log a y, then x = y

4. NATURAL LOGARITHMIC FUNCTION The natural logarithmic function is f(x) = log e x = ln x where x > 0.

5. SPECIAL PROPERTIES OF NATURAL LOG, ln 1. ln 1 = 0 because e 0 = 1. 2. ln e = 1 because e 1 = e. 3. ln e x = x because e x = e x 4. If ln x = ln y, then x = y

6. EXAMPLE: MORTGAGE PAYMENT The length, t years, of a $150,000- mortgage at 10% with monthly payments of $x (>$1250) is approximated by:

7. FORMULA DERIVATION Solving for t yields: P = monthly payment A = amount of loan r = rate (%), n = # of payments/yr t years

8. FORMULA DERIVATION Solving for t yields the formula below: P = monthly payment A = amount of loan r = rate (%), n = # of payments/yr t years

9. EXAMPLE continued: MORTGAGE PAYMENT How many years are needed with monthly payments of $1500? Answer is: t =17.99 or 18 years. (Total paid $323,871.27.)

10. LET US WORK EXAMPLES.