Introduction to Logarithms
Warm Up A country’s population is 3 million and increases by 5% every year. That same country’s food supply is capable of feeding 6 million and increases by 200,000 every year. Q1: Write a function that models the population. Q2: Write a function that models the food supply. Q3: Will the country always produce enough food to feed all of its population? Explain. Q4: Find how many years it takes for the population to begin to exceed the amount of food available. Introduction to Logarithmic Functions Definition and Evaluation of Logarithm Exponential form
10! = 10,000
Logarithmic form
4 = log!" 10,000
exponent
base
Let b be a positive where 𝑏 ≠ 1 . The logarithm to the base b, denoted log ! 𝑥 is defined as follows log ! 𝑥 = 𝑦 if and only if 𝑏 ! = 𝑥 Example Writing in Logarithmic Form Change the following from exponential form to logarithmic form ! !
!
1a) 16 = 4! 1b) 243 = 3! 1c) 2a) 6!.! = 𝑘 2b) 𝑎! = 34 2c) 8!.! = 𝑙 !
= !" Example Evaluating Logarithms Write the following in exponential form then evaluate 3a) log ! 8 3b)log ! 16 !"
3c)log ! !" !
FREEZE! Logarithm War Time Properties of Logarithm Complete columns 1 – 3 first then make a conjecture. log ! 𝑚 log ! 𝑛 log!" 10 = ____ log!" 1000 = ____ log ! 8 = ____ log ! 4 = ____ log ! 27 = ____ log ! 9 = ____ log ! (𝑚 ∙ 𝑛) log!" 10 ∙ 1000 = ____ log ! 8 ∙ 4 = ____ log ! 27 ∙ 9 = ____ log !
𝑚
𝑛
10
= ____ 1000
8
log !
= ____ 4
27
log !
= _____ 9
log!"
Complete the following table and discuss the results. log ! 𝑚 ! 𝑟log ! 𝑚 log ! 𝑚 ! ________𝑟log ! 𝑚 log!" 10! = ____ 3log!" 10 = ____ log!" 10! ______3log!" 10 log ! 4! = ____ 2log ! 4 = ____ log ! 4! ______2log ! 4 Discussion (Properties of logarithm) a) What conjecture can you make about the relationship between log ! 𝑚, log ! 𝑛, and log ! 𝑚 ∙ 𝑛 ? !
b) What conjecture can you make about the relationship between log ! 𝑚, log ! 𝑛, and log ! ! ? c) What conjecture can you make about the relationship between log ! 𝑚 ! and 𝑟 log ! 𝑚? Product Property log ! 𝑀𝑁 = Quotient Property 𝑀
log ! = 𝑁
Power Property !
log ! 𝑀 = Example Writing Properties and Combining as a Single Logarithm State the property or properties and rewrite each expression as a single log. 1a) log ! 8 + log ! 5 1b) log ! 27 − log ! 3 1c) 2log ! 𝑥 + 3log ! 𝑦 1d) 3log ! 2 − 2log ! 5 1e) 2log ! 8 − log ! 4 + 3log ! 𝑥 Example Writing Properties and Expanding Logarithms State the property or properties and expand each logarithm 2a) log ! (𝑥𝑦) 2b) log ! 𝑥 ! 𝑦𝑧 ! 2d) log !
(!")!
!!
2e) log !
!" !
!
2c) log ! [ 2𝑚 ! 𝑙] Example Applications of Logarithm Properties The energy E (in kilocalories per gram molecule) required to transport a substance from the outside to the inside of a living cell is given by the equation, 𝐸 = 1.4 log 𝐶! ) − log(𝐶! , where 𝐶! is the concentrate of the substance outside the cell and 𝐶! is the concentrate inside the cell. 1. Write the value of E as a single logarithm. 2. If the concentration of a particular substance inside a cell is three times the concentration outside the cell, how much energy is required to transport the substance from the outside to inside the cell? Homework Complete the statement to validate why the following is true. 1) log ! 1 = 0 since written in exponential form ______________________. 2) log ! 𝑎 = 1 since written in exponential form ______________________. 3) log ! 𝑎 ! = 𝑥 since written in exponential form _____________________. Write each equation in logarithmic form. 5) 70 = 1 4) 53 = 125 6) 34 = 81 Write each equation in exponential form. 8) log2 64 = 6 7) log6 216 = 3 9)
Evaluate each expression. 10) 13) 12) log10 0.0001 11) log3 81 14) 15) log9 1 17) log9 9(n + 1) 16) 18) 2log2 32 Solve each equation. Check your solutions. 19) log10 n = -­‐3 20) 21) Determine if the following is true or false. Justify your answer. If false, provide a counter example. !
22.) log ! 9𝑥
23.) log !
! !
!
= 2log ! 9 + log ! 𝑥 = 3log ! 𝑥 − 3log ! 𝑧 24.) log ! 25𝑥 ! = 2 + 4log ! 𝑥 Write each logarithmic expression as a single logarithm. 25.) log ! 20 − log ! 4 26.) 3log ! 𝑥 + log ! 𝑦 Expand each expression using the properties of logarithms. 27.) log !
!!
!
Use the properties of logarithms to solve each equation. 29.) log6 2c + log6 8 = log6 80 30.) log4 x – 3 log4 2 = 2 28.) log ! 𝑎! 𝑏 !