Exponential functions Logarithmic functions 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

Solve for x: To solve exponential equations, get the bases equal. One to one property! Bases must be the same Solve for x:

BONUS!!

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 5½% 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

Natural log-base e 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 e is an irrational number like π. e = 2.718... If we use a natural log, we indicate by writing ln instead of log and no base is needed. 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 Evaluate using Change of Base log6 8 log3 12

Rewrite as a exponential equation: log4 16 = 2 log3 729 = 6 log8 512 = 3 log16 8 = 3/4

Rewrite as a logarithm: 43 = 64 1251/3 = 5 113 = 1331 163/4 = 8

To find the exact value of a logarithm (or evaluate), we can change the equation to an exponential one. Evaluate: log3 81 log1/2 256

Evaluate: log13 169

8.4 Solving logarithmic equations

Solve: Change to exponential form

log6x = log69

log3(x2 - 15) = log3 2x log4(5x-4) > log43x

Solve Base e Equations Good to know ln ex = 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 e4x - 12 = 15

Solve Natural Logs Good to know: eln x = x 5 ln 6x = 8 3 ln 4x = 24

Continuously Compound Interest A = Pert 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 logx ab= logx a + logx b Quotient Property logx a/b = logx a - logx b Power of Properties logb Ax = xlogb A

Simplify: Expand:

Use log4 2 = .5 to approximate log4 32

Use log37 = 1.77 to approximate log3 49

Solve

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 = log10(1/H+). Find the amount of hydrogen in a liter of acid rain that has a pH of 4.2.

Write in log form: ex = 9 e7 = x Write in exponential form: ln x = 2.143 ln 18 = x

Simplify the expression: 6 ln 8 - 2 ln 4 2 ln 5 + 4 ln 2 + ln 5y

To solve exponential equations, get the bases equal. we can’t get the bases equal here.

Solve: 6x = 42

Key Chapter points: Exponential functions and graphs Logarithmic functions Properties of logarithms Exponential and Logarithmic equations