Logarithmic Functions - Answer Key

Today we did a few examples from the class booklet on Log Functions

Example 1: a) v = 3      b) w = 4     c)  x = -4       d) y = 0.83     e)  z = -3
Example 2:  a)  v = 4    b)  w = 2    c)  x = -3       d)  y = 7.39      e)  z = 3
Example 3:  a)  6         b)   3      c)   -2    d)    2      e) 2              f)   12
Example 4: B (Dividing entire logs is not the same as dividing the arguments)
Example 5:  NR x = 32
Example 6:  D
Example 7:  NR x = 1.29
Example 8:  NR x = 4.46

Example 9:  C Logarithmic because the independent variable and dependent variable are interchanged
Enter L1 and L2, Stat Calc 9 (Ln Regression)
y = 32434.05 + -6974.1 lnx

y=  VARS    #5 Statistics > > EQ Enter

Graph with x:{0, 120, 10} & Y: {0, 16000, 1000} for window settings

Example 10
Graph y = 8848 and find intersection at X = 90
Graph y = 1049 and find intersection at X = 30
Subtract to find the difference = 60

Evaluating Logs using Exponents and Using Log Laws to Evaluate Values

Today we looked at the similarities between exponent forms and Logarithmic fomrs of values of numbers.

We saw that for y = b^x we can rearrange this exponential form to a Logarithmic form such that

y = b^x < -> logbx = y

We can use this to evaluate expontial values

It is easy to see that 10^x = 1000 because 10^3 = 1000, so 10^x = 10^3, and dropping bases x = 3

But what if 10^x = 81?  This means: 10 to the power of what will result in 81?

We can see that 

10^1 = 10 and 10^2 = 100

81 lies between 10 and 100 (closer to 100) therefore the exponent of 10 must be between 1 & 2 (and closer to 2)

81 = 10^x -> changing exponents to log form x = log 1081,  x = 1.908.

This means 10^1.908  = 81

We summarized the steps to be:

• REARRANGE Log form to Exponential form
• re-write exponential forms to have the SAME BASE
• Use EXPONENT LAWS if necessary
• DROP BASES due to equivalent forms of exponents
• SOLVE

We also discovered the laws that apply to log functions are similar to exponent laws.

See the notes below.

• Apply LOG LAWS where you can to create a single Log function
• REARRANGE Log form to Exponential form
• re-write exponential forms to have the SAME BASE
• Use EXPONENT LAWS if necessary
• DROP BASES due to equivalent forms of exponents
• SOLVE

HW - see calendar from the first Post for today's HW.

Intro to Logarithmic Functions

Logarithmic functions are the inverse (reverse the positions of x and y) of exponential functions.

Using the Log of a exponent allows us to solve exponential equations algebraically instead of graphically (more on this to come later).

We

• investigated how Logarithmic functions compared to Exponential functions
• compared the Domain and Range
• compared the x and y intercepts
• compared what the a value represents in each situation
• exponential functions: a is the initial value
• logarithmic functions: a is the value of the function when x = 10 ( or when x= the value of the base of the log - more to come on this later)

Homework

Lots of homework - check the HW section for July 17 on the first blog post in June

Exponential Equations Notes

We did a lot of work on Exponential Functions
We applied the General y = a x b ^x function to a number of different situations

Recall that in all of these the common ideas of

a = the initial value:

• maybe for bacteria it is the initial number of bacteria, 
• for financial situations the Principal (or initial) investment

b = the rate of growth or change:

• doubling b = 2
• growth by % in financial situations b = 1 + i, where i is the interest rate/100
• growth by % in financial situations where there are compounding periods 
• divide the annual interest rate by the number of compounds per year

x = the number of times the event occurs 

• bacterial growth hours or 
• total number of compounds  = years x # compounds per year

There is a lot of homework

Check the HW chart on the first Blog post from June

Intro to Exponential Functions

There are a number of different functions
Linear - y= mx+b create a straight function, where m is the slope and (0,b) is the y-intercept

Quadratic - y = ax^2 + bx + c creates a U shaped function, were a represents whether or not the function opens up (+a) or opens down (-a) with a y-intercept at (0,c)

The Exponential functions are those where x is the exponent of the function y = a b^x

We saw that with positive exponents

• a is the y-intercept 
• when a is positive the function behaves as normal
• when a is negative, the function is the reflection of the function when a is positive
• the b value determines the behaviour of the functions - either going up or going down
• when b > 0, the function increases from left to right
• when 0 < b < 1, (proper fractions or decimals) the function decreases from left to right

See the examples below

HW

P337 # 1-3

P346 # 1-7, 11, 15, 18

QUICK CHECK UP ON THIS TOPIC ON THURSDAY!!!

Today we worked on problems where we have to decide either Permutations or Combinations to solve the problem.
Good examples were
P122 Example #1 and P152 Example #1
Here is a flowchart to help you to decide when to use Perms and when to use combs.

HW 

P126 # 1,2,5,6,7,10

and

P159 #2-10, 12

Combinations - when order does NOT matter

Combinations are ways of arranging elements in a manner that the order does not matter

HW P110 #1,2 & P 118 Any 8 of 1-12