Write your questions and
thoughts here!
7.2 – Logarithmic Functions Name:____________________
1
𝑦 = log ! 𝑥 if and only if 𝑏 ! = 𝑥
Definition of Logarithm: 𝑏 > 0, 𝑏 ≠ 1, 𝑥 > 0 How does Mr. Bean say log ! 𝑥 ? How does Mr. Kelly say log ! 𝑥 in his head? !
Recall: An ____________ function is of the form 𝑦 = 𝑏 A __________________ function is of the form 𝑥 = 𝑏! Rewrite in exponential form: These functions are inverse functions! Rewrite in logarithmic form: 1. log ! 125 = 3
2. !
!
!
!
= !
Evaluate the following logarithms (without a calculator): 3.
4. log !
log ! 8
!
5.
!"
log ! 1
PROPERTIES OF LOGARITHMS Product Property: Quotient Property: !
log ! 𝑥𝑦 = log ! 𝑥 + log ! 𝑦
Expand: 6.
log ! = log ! 𝑥 − log ! 𝑦
!
Power Property: log ! 𝑥 ! = 𝑦log ! 𝑥 EXPANDING AND CONDENSING LOGRITHMS:
Condense:
5
𝑝
log4 3𝑥5 𝑦
7.
log3 4
8. 2 log4 8 − log4 5 + 0.5 log4 25
𝑞
Method: CHANGE-­‐OF-­‐BASE: 𝐥𝐨𝐠 𝑪 𝒂 =
9.
10.
𝐥𝐨𝐠 𝒃 𝒂
𝐥𝐨𝐠 𝒃 𝒄
Solve for x: 11.
41 = 7 !
12.
0.25 = 2 !
Evaluate log!" 5
Evaluate log ! 20
Bean:
(take log both sides) Brust:
(cancel by using
log of base)
Sully:
JUST GRAPH IT, BABY!
Write your questions and
thoughts here!
7.2 – Logarithmic Functions Other Reminders: log ! 𝑏
= log ! 1 = log!" 𝑥 is called: log ! 𝑏 ! = 𝑏 !"#! ! = log ! 𝑥 is called: Solve for the unknown variable. 14. log 𝑥 = 3.0876 15. ln 𝑥 = −0.9128 GRAPHING LOGARITHMIC FUNCTIONS: Use the properties of logarithms to find the inverse of the given function. (Hint! Switch x and y and solve for y!)
16. 𝑓(𝑥) = 4 ! 17. 𝑓 (𝑥) = 3!! ∑ ℝ𝒆
18. 19. 𝑓 𝑥 = log 𝑥 + 1 20. ln 𝑦 = 2 ln 𝑥 𝑓(𝑥) = 5
!
!
!
7.2 Practice – LOGARITHMIC FUNCTIONS
Name: _____________________________ Pre-­‐Calculus You might as well get these bad boys out of the way first. Solve for each unknown variable. 2. 3. Quick Review 1. For 4-­‐6, Expand the logarithm. (NOT L I K E T H I S ! ! !) !!
4. 𝑙𝑜𝑔! 𝑎𝑏 ! 𝑐
5.
ln !
6. log
!
!" !
!
For 4-­‐6, Rewrite the expression as a single log. (C o n d e n se!) !"# !
7. log a – 2 log b + 3 log c
8. 2 ln x + 5 ln y – ln z3
9. 𝑙𝑜𝑔! 𝑦 + 7𝑙𝑜𝑔! 𝑥 + !
!
Solve for x using the “Bean method” (change of base formula). Show your work! Go out four places! 10. 4! = 14
11. 8! = 12
12. 100! = 1000
Solve for x by using the “Brust method” (canceling the base with logs). Show your work! Go out four places! 13. 5! = 15
14. 4! = 1024
15. 100! = 50
Solve for x by using the Sully method (by graphing). Tell the point of intersection used to solve the equation. 16. 3! = 13
17.
15! = 4
18. 100! = 10
x = ________ Point ( , )
x = ________ Point ( , )
x = ________ Point ( , )
Find x, y, or b as indicated in the following problems. 19. 𝑙𝑜𝑔! 𝑥 = 2
20. 𝑙𝑜𝑔!" 8 = 𝑦
21.
23.
𝑙𝑜𝑔! 𝑥 =
!
24.
!
𝑙𝑜𝑔! 16 = 2
𝑙𝑜𝑔! 9 = 𝑦
22.
25.
!
𝑙𝑜𝑔! 1 = 0
𝑙𝑜𝑔! 1000 =
Use logarithms to find the inverse of the given function.
26. 𝑓 𝑥 = 6!
27. 𝑓 𝑥 = 3! + 4
28. 𝑓 𝑥 = 3!!!
29. 𝑓 𝑥 = ln (3𝑥)
31*. log 𝑦 =
30. log 𝑦 = 3 log 𝑥 + 4
* 32. Condense into a single logarithm.
!"#! !
!!! !
− 𝑙𝑜𝑔! 𝑦 ! + !
!
!
32. Expand.
x3 y
ln
z
!"# !!!
!
!
!
7.2 Application and Extension 1.
Expand: log
!
!
2.
!!
Solve for x: 3x = 99
3.
When Sully is ready to retire, he has plans of moving to New York City to become a butcher. In fact, he
wants to open his own butcher shop, “The New York Metzgerei,” where he can sell his signature product: Sullamy Picante! Sully has to cook the meat and then let it cool while recording the temperature during the production process. One day, Sully observes the following temperatures: Time (min) 10 Temperature (degrees 51 above room temp in F) 14 20 22 26 30 36 40 42 44 41 30 26 21 17 11 8 6 5 Diff from Room Temp (oF) a. Plot the data on the graph to the right. Enter
Time into L1 and Temp into L2..
b. Would a linear model be appropriate for this data?
Why or why not?
c. To “straighten” the data, take the common log
of each of the temperatures. (Log L2 à L3)
10o 0o
Complete the table:
Time (min) 10 14 20 22 26 30 36 10
Time (Min) 40 42 Log (Temp) d. Calculate the linear regression for Time vs. Log Diff Temp.
Plot the data and graph line of best fit on the graph. (For
Log Temp (oF) help with your calculator, watch the Application Help Video!)
e. Is the data straighter?
1.0 10
(Flip it like an Algebro!)
Flip it like an Algebro
Time (Min)
44 f. Now complete a LINEAR REGRESSION using your calculator on Time and Log Temp. Write your linear
equation below, accurate to 4 decimal places. Remember, we aren’t using y, we are using log y.
(Use LinReg L1, L3) a = __________________ b = _________________ Log y = ax + b Log y = ________________________________________ g. In statistics, straighter data leads to more accurate predictions. We take the log of the dependent
variable to straighten out exponential data. But you know what? We like the ORIGINAL variable and
we hate equations with logs in them. Let’s use our Log rules to reverse transform the equation into an
exponential equation. For even more fun, let’s bust out the 2 column proof:
Regression Equation Statements 1.
Log y = ___________________________
Reasons 1.
Given
Your equation from above!
2. 2. Raise 10 to the power of each side of the equation.
3. 3. Write the exponent sum as the product of a common base.
4. 4. Compute 10
5. 5. Rewrite the exponent product as a power to a power.
6. 6. Compute 10
7. 7. Rewrite your equation in the form y = ab
y-­‐intercept value
slope of equation
.
x
h. Confirm your exponential regression equation with ExpReg in your calculator. (Statà Calcà
ExpReg L1, L2.) Be sure you use the original data and not the transform data in L3. Congrats! You just
learned how the calculator calculates an exponential regression… call it “The Great Regression of 7.2!”
i.
How hot was the Sullamy Picante when Sully took it out of the oven? (Assume a room temperature of
72o) Use your equation to figure it out. Show your work below.