Which plan yields the most interest. Invest $100 Plan A: A 7 Which plan yields the most interest? Invest $100 Plan A: A 7.5% annual rate compounded monthly for 4 years Plan B: A 7.2% annual rate compounded daily for 4 years Plan C: A 7% annual rate compounded continuously for 4 years Warm Up Solve: −160=−5 𝑎+16 5 3 Solve: 3𝑥−2 +1= 5𝑥−1

Plan A: A 7.5% annual rate compounded monthly for 4 years Which plan yields the most interest? Invest $100 Plan A: A 7.5% annual rate compounded monthly for 4 years Plan B: A 7.2% annual rate compounded daily for 4 years Plan C: A 7% annual rate compounded continuously for 4 years 𝟏𝟎𝟎 (𝟏+ .𝟎𝟕𝟓 𝟏𝟐 ) 𝟏𝟐∙𝟒 =𝟏𝟑𝟒.𝟖𝟔 𝟏𝟎𝟎 (𝟏+ .𝟎𝟕𝟐 𝟑𝟔𝟓 ) 𝟑𝟔𝟓∙𝟒 =𝟏𝟑𝟑.𝟑𝟕 𝟏𝟎𝟎 𝒆 .𝟎𝟕∙𝟒 =𝟏𝟑𝟐.𝟑𝟏

5.5 Logarithmic Functions Graph: 𝒚= 𝟐 𝒙 Graph the inverse. Can you find the inverse equation algebraically? 𝒙= 𝟐 𝒚 The inverse equation is 𝒚= 𝒍𝒐𝒈 𝟐 𝒙

Logs vs exponential functions Logarithms are the inverses of exponential functions. The output,or answer, to a log expression is an exponent

5.5 Logarithmic Functions Changing from exponential to log form and vice versa: 𝑙𝑜𝑔 𝑏 𝑥=𝑎 ↔ 𝑏 𝑎 =𝑥 Example: If 𝟐 𝟑 =𝟖 then 𝒍𝒐𝒈 𝟐 𝟖=𝟑

𝑙𝑜𝑔 𝑏 𝑥=𝑎 ↔ 𝑏 𝑎 =𝑥 a. 𝑙𝑜𝑔 12 1=0 a. 52=25 b. 2-3=1/8 b. 𝑙𝑜𝑔 27 3= 1 3 𝑙𝑜𝑔 𝑏 𝑥=𝑎 ↔ 𝑏 𝑎 =𝑥 Write each in exponential form. a. 𝑙𝑜𝑔 12 1=0 b. 𝑙𝑜𝑔 27 3= 1 3 Write each in log form. a. 52=25 b. 2-3=1/8 𝒍𝒐𝒈 𝟓 𝟐𝟓=𝟐 𝟏𝟐 𝟎 =𝟏 𝒍𝒐𝒈 𝟐 𝟏 𝟖 =−𝟑 𝟐𝟕 𝟏 𝟑 =𝟑

Strategy for evaluating logs without at calculator Step 1: switch to exponent form Step 2: what exponent makes the statement true? Answer step two by changing to like bases, if necessary.

Evaluate without a calculator: =𝑥 𝒙=𝟐 𝟏𝟐 𝒙 =𝟏𝟒𝟒 𝑙𝑜𝑔 12 144 𝑙𝑜𝑔 10 0.001 𝑙𝑜𝑔 5 5 =𝑥 𝟏𝟎 𝒙 =.𝟎𝟎𝟏 𝒙=−𝟑 𝒙= 𝟏 𝟐 =𝑥 𝟓 𝒙 = 𝟓 Hint: set the expression equal to x. Rewrite in exponential form and solve.

Problem 2 𝑙𝑜𝑔 10 0.001=𝑥 𝟏𝟎 𝒙 =.𝟎𝟎𝟏→ 𝟏𝟎 𝒙 = 𝟏 𝟏𝟎𝟎𝟎 solve by changing to like bases: 10 𝑥 = 1 10 3 → 10 𝑥 = 10 −3 →𝑠𝑜 𝑥=−3 So 𝑙𝑜𝑔 10 0.001=−3

Problem 3 𝑙𝑜𝑔 5 5 =𝑥→ 𝟓 𝒙 = 𝟓 Reminder: 𝑥 = 𝑥 1 2 𝑙𝑜𝑔 5 5 =𝑥→ 𝟓 𝒙 = 𝟓 Reminder: 𝑥 = 𝑥 1 2 𝟓 𝒙 = 𝟓 → 𝟓 𝒙 = 𝟓 𝟏/𝟐 →𝒙= 𝟏 𝟐

The Common Log Base 10 Written as log(x) A logarithm can have any positive value as its base, but two log bases are more useful than the others. The Common Log Base 10 Written as log(x) Base 10 is the default for logs The log key on your calculator has base 10 Examples: pH (the measure of a substance's acidity or alkalinity), decibels (the measure of sound intensity), the Richter scale (the measure of earthquake intensity)

𝒍 𝒏 𝒙 =𝒌 𝒆 𝒌 =𝒙 The Natural Log The other base that is used often is e 𝒍𝒐𝒈𝒆 𝒙 is usually written as 𝒍𝒏 𝒙 The natural log key on your calculator is LN Just as the number e arises naturally in math and science, so does the natural log 𝒍 𝒏 𝒙 =𝒌 𝒆 𝒌 =𝒙

Find the value to the nearest hundredth: a. 10 𝑥 =75 b. 𝑒 𝑥 =75 𝒙=𝒍𝒐𝒈𝟕𝟓 ≈𝟏.𝟖𝟖 𝒙=𝒍𝒏𝟕𝟓≈𝟒.𝟑𝟐

Solve Without a Calculator you may leave your answer in terms of e 𝑙𝑜𝑔𝑥=4 𝑙𝑛𝑥=2 𝑙𝑜𝑔 5 𝑥=2 𝑙𝑜𝑔 𝑥 121=2 𝑙𝑜𝑔 𝑥 1 2 =−1 10,000 𝑒 2 25 11 2

Homework due Friday P194 1-21 odd, plus 35-43 odd OR 19-43odd