WARM - UP Evaluate: log 3 81 Solve for x: log5 (2x+3) = log5 (4x -3) Graph: log 2 (𝑥+1) −3
Change of Base Formula, Expanding & Condensing Logarithms
OBJECTIVES Students will be able to... Use logarithmic Properties for condensing and expanding logarithmic expressions Evaluate logs and natural logs Use the change of base formula to evaluate logs with bases other than 10
HOMEWORK Worksheets “The Change of Base Formula” “Properties of Logarithms” – Expanding and Condensing
Change of Base Formula Let a, b and x be positive real numbers such that 𝑎≠1 and b≠1. Then log 𝑎 𝑥 can be converted to a different base as follows. Base b Base 10 Base e log 𝑎 𝑥 = log 𝑏 𝑥 log 𝑎 𝑥 log 𝑎 𝑥 = log 𝑥 log 𝑎 log 𝑎 𝑥 = ln 𝑥 ln 𝑎
EXAMPLES With logarithms: 1) log 4 25 2) log 2 12 With natural logs:
practice “Change of Base Formula” Worksheet ___________ minutes; stopping at __________
Properties of logarithms Let a be a positive number such that 𝑎≠1, and let n be a real number. If u and v are positive real numbers, the following properties are true: Log w/ Base a Natural Log Product Property: Quotient Property: Power Property: log 𝑎 𝑢𝑣 = log 𝑎 𝑢 + log 𝑎 𝑣 ln 𝑢𝑣 = ln 𝑢 + ln 𝑣 log 𝑎 𝑢 𝑣 = log 𝑎 𝑢 − log 𝑎 𝑣 ln 𝑢 𝑣 = ln 𝑢 − ln 𝑣 log 𝑎 𝑢 𝑛 = 𝑛 log 𝑎 𝑢 ln 𝑢 𝑛 =𝑛 ln 𝑢
Examples Write each logarithm in terms of ln 2 and ln 3 1) ln 6 2) ln 2 27
examples Find the exact value of each expression without using a calculator 1) log 5 3 5 2) ln 𝑒 6 − ln 𝑒 2
Rewriting logarithmic expressions Expanding Logarithmic Functions Examples: 1) log 4 5 𝑥 3 𝑦 2) ln 3𝑥−5 7
Rewriting logarithmic expressions Condensing Logarithmic Functions Examples: 1) 1 2 log 𝑥 +3 log (𝑥+1) 2) 2 ln 𝑥+2 − ln 𝑥 3) 1 3 [ log 2 𝑥 + log 2 𝑥+1 ]
practice “Properties of Logarithms” Worksheet ___________ minutes; stopping at __________
CLOSURE On the note card provided: (Hand in before walking out the door!) 1) Evaluate the logarithm using the change of base formula: log 15 1250 2) Rewrite and simplify: ln (5 𝑒 6 ) 3) Expand the logarithm: log 10 𝑦 2 4) Condense the logarithms: log 5 8 − log 5 𝑡