Property #1 – Product Property log a (cd) = Algebra 2/Trigonometry Name: __________________________ Section 8.5 Fill In Notes Date: ___________________________ 8.5 Properties of Logarithms Most calculators only have two types of log keys: 1.) Common Logarithms (Base ____) 2.) Natural Logarithms (Base ____) To evaluate logarithms to other bases we need to use the CHANGE OF BASE FORMULA! log a x can be converted to a different base as follows: BASE b BASE 10 BASE e log a x = Changing Bases using Common & Natural Logarithms Common Log Natural Log log 4 25 = log 2 12 = log 3 16 = log 5 22 = Property #1 – Product Property log a (cd) = Log Properties: How well do you actually know them? log 3 3 =___ Examples: log 5 125 = ln (xyz) =

Property #2 – Quotient Property log a c d = Log Properties: How well do you actually know them? log 7 1 =___ Examples: log 4 1 64 = log xy zw = Property #3 – Power Property log a c w = Log Properties: How well do you actually know them? ln e =___ Examples: log 5 7 12 = log 7 4 7 = log 4 4 10 = ln56= Simplify each expression. log 5 75 − log 5 3 ln 𝑒 6 𝑒 2 log 4 2 + log 4 32

Condense each expression into one logarithm. Log Properties: How well do you actually know them? log 104=___ Condense each expression into one logarithm. 1.) 1 2 log 𝑥+3log⁡(𝑥+1) 2.) 2 ln(x+2) – lnx 3.) log 3 (𝑥+2) − log 3 𝑥 −2 log 3 𝑥 Log Properties: How well do you actually know them? ln e=___ Expand each expression. 1.) log 4 5 𝑥 3 𝑦 2.) ln 3𝑥−5 7 3.) log(4x3y5z2)

Algebra 2/Trigonometry Name: __________________________ Section 8.6 Fill In Notes Date: ___________________________ All of these functions are one-to-one. (They pass the ________________________). This means that every x value has _______________________________. Solving Exponential Equations Using the One-to-One Property: Example 1: Example 2: Example 3: Example 4: Solving Exponential Equations w/o One-to-One: 5. 3x = 30 6. (¼)x = 60 7. 3(2x) = 42 8. ex = 7 9. 5 - 3ex = 2 10. 2(32t-5) - 4 = 11 Example 11 e2x - 7ex = -12 Example 12 e2x - 4ex – 5 = 0

Solving Logarithmic Equations: Recall: If _____________________________________________ . Solving Logarithmic Equations Using the One-to-One Property: Example 1: Example 2: Example 3: Solving Logarithmic Equations w/o One-to-One: 4) lnx = -3 5) logx = -1 6) 2log5(3x) = 4 7) lnx – ln3 = 0 8) log3(2x + 1) + log3(2) = log3(5x) 9) log6(3x + 14) – log6(5) = log6(2x) 10) log(5x) + log(x – 1) = 2 11) log4x – log4(x – 1) = ½

Algebra 2/Trigonometry Name: __________________________ Section 8.6 Fill In Notes Date: ___________________________ Solving Logarithmic Equations, continued: logx + log(x + 4) = 1 Application Example 1: If you had $2000 to invest in an account with an APR of 6% (compounded monthly), how many years would it take for the account to be worth $5000? Round to the nearest hundredths of a year. Application Example 2: You have deposited $500 in an account that pays 6.75% interest, compounded continuously. How long will it take your money to double? Round to the nearest hundredths of a year.