Text of Unit 7 Logarithms Exponential functions Logarithmic functions Using properties of logarithms...
Unit 7 Logarithms Exponential functions Logarithmic functions Using properties of logarithms Exponential and Logarithmic equations Exponential and logarithmic models
8.2 Solving exponential equations and inequalities
To solve exponential equations, get the bases equal. Solve for x: One to one property! Bases must be the same
Compound Interest Formulas After t years, the balance A in an account with principal P and annual interest rate r (in decimal form) is given by the following formula: 1) For n compounding per year: Find the account balance after 20 years if $100 is placed in an account that pays 1.2% interest compounded twice a month.
If $350,000 is invested at a rate of 5% per year, find the amount of the investment at the end of 10 years for the following compounding methods: a) Quarterly b) Monthly
Solving exponential inequalities is similar to equations, make sure the bases are equal. Solve:
8.3 Logarithmic Functions Write exponential functions as logarithms Write logarithmic functions as exponential functions
A logarithm function is another way to write an exponential function y is the logarithm, a is the base, x is the number
Common log-base 10 When we use a common log with base 10, it is not necessary to indicate the base. Use the log button on the calculator to take use base 10 log of any number. Natural log-base e If we use a natural log, we indicate by writing ln instead of log and no base is needed. e is an irrational number like . e = 2.718... Use the ln button on the calculator to take the natural log of any number.
Evaluate using Change of Base How do we evaluate logarithms that are not common? Change of base formula log 6 8 log 3 12
Solve Base e Equations Good to know ln e x = x 4 e -2x - 5 = 3 Add 5 to both sides Divide 4 to both sides Take ln of both sides New property
Solve: 3 e 4x - 12 = 15
Good to know: e ln x = x Solve Natural Logs 3 ln 4x = 24 5 ln 6x = 8
Continuously Compound Interest A = Pe rt Joan was born and her parents deposited $2000 into a college savings account paying 4% interest compounded continuously. What would be the balance after 15 years.
8.5 Using Properties of logarithms Rewrite logarithms with a different base Use properties of logarithms to evaluate or rewrite logarithmic expressions Use properties of logarithms to expand or condense logarithmic expressions Use logarithmic functions to model real-life problems
Product Property log x ab= log x a + log x b Quotient Property log x a/b = log x a - log x b Power of Properties log b A x = xlog b A
Use log 4 2 =.5 to approximate log 4 32 Use log 4 3 =.7925 to approximate log 4 192 Use log 5 2 =.4307 to approximate log 5 250
Use log 3 7 = 1.77 to approximate log 3 49 Use log 5 6 = 1.11 to approximate log 5 216
Real World Applications The Ph of a substance is defined as the concentration of hydrogen ions [H + ] in moles. It is given by the formula pH = log 10 (1/H + ). Find the amount of hydrogen in a liter of acid rain that has a pH of 4.2.
Write in log form: e x = 9 e 7 = x Write in exponential form: ln x = 2.143 ln 18 = x