5.7 – Exponential Equations

5.7 Exponential Equations Objectives: I will be able to…  Solve Exponential Equations using the Change of Base Formula Vocabulary: logarithms, natural logarithms

Daily Objectives Perform the Change of Base formula. Master solving tricky logarithm equations. ▫Exponent variables

Topic One: Change of Base Formula The BASE goes to the BOTTOM We typically use log base 10 so we can solve on our calculator!

Example 1: Using Change of Base Formula This is a great problem to use the change of base formula on – Why? On the other hand, why is problem #2 not the type of problem that you should use the change of base formula on? #3 can work with either – Why?

Example 2: Solve for a variable exponent when it’s impossible to get same base 1.Get the base/exponent by itself on one side of the equation (use PEMDAS) 2.Take the log of both sides 3.Bring down the exponent via the Log Power Rule 4.Get x by itself using Change of Base Rule! 5.**Remember: ln(e)=1 and log(10)=1 x ~

Example 3: Solve for a variable exponent x ~

t ~ years Example 4: Solve for a variable exponent (time)

Practice time!

#1: Solve for a variable exponent x ~

#2: Solve for a variable exponent y ~ 2.322

#3: Solve for a variable exponent 5 = (1.6) x x ~

#4: Solve for a variable exponent x ~

#5: Solve for a variable exponent x ~

#6: Solve for a variable exponent No solution

#7: Solve for a variable exponent The number of bacteria present in a culture N(t) at time t hours is given by N(t)=3000(2) t How long will it take for the population to triple in size? t ~ hours

Homework p. 205 #5-14

t ~ years Example 7: Solve for a variable exponent (time)

Example 8: Solve for a variable exponent A biologist decides that an epidemic spreads through a population of a city according to the following model p(t) = 1 − e −0.34t where p(t) represents that fraction of the city’s population which has come down with the disease, and t is in weeks. How long will it take for 90% of the city to become infected? t ~ weeks

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