Inverse functions and Logarithms

Inverse functions and Logarithms

How do Logarithms and exponentials connect?

Multiple Choice

Practice! Contract this logarithm expression into a single logarithm, simplifying as much as possible:

 $\log42-\left(\log7+\log2\right)$  

•  $\log\left(12\right)$  

•  $\log\ \left(\frac{14}{3}\right)$  

•  $\log\left(3\right)$  

•  $\log\left(33\right)$  

Multiple Choice

Fully expand this logarithm expression:

 $\log\left(\frac{3x}{y^2}\right)$  

•  $x\log\left(3\right)-2\log\left(y\right)$  

•  $\log\left(3x\right)-\log\left(y^2\right)$  

•  $\log\left(x\right)+\log\left(3\right)-2\log\left(y\right)$  

•  $2\left(\log\left(3\right)+\log\left(x\right)-\log\left(y\right)\right)$  

What is an inverse?

Inverse functions are functions that allow you to UNDO something that a function does.

For instance, say a function we have is f(x) = 3x.
f(1) = 3•1 = 3.     f(5) = 3•5 = 15
The inverse function will undo what the regular function does. 

• In this case, the inverse function would be 

 $f^{-1}\left(x\right)=\frac{x}{3}$  
Steinkamp shows some examples on iPad - what goes in and out!

Switching inputs and outputs

Open Ended

A function f(x) takes in x = 2, and gives an output of 4.

f(2) = 4

f(x) has an inverse function,

 $f^{-1}\left(x\right)$  .
Can we tell what will  $f^{-1}\left(4\right)$   give as an output?

If we can, what will it be and why?

Fill in the Blank

The y intercept of a graph of f(x) is (0, 4). What point do we know must be on the graph of the inverse?

Most functions have inverses!

• The inverse of $f\left(x\right)\ =\ x^2$   is ...

• (iPad examples, including  with multiple steps)

What do inverses look like?

Desmos demonstration

https://www.desmos.com/calculator/ntbh9sdd43

Graphs of inverses: The Rule!

The inverse of a function will always look the same as the original function Flipped over the line y = x.

This also means that any coordinate on the original function (a, b), will have a partner point (b, a) on the inverse graph. (see desmos)

Multiple Select

Which graphs show a pair of a graph and its inverse?

What does this have to do with exponential functions and logarithms?

• Desmos: Graph of y =10^x and y = log (x)

log(x) is the inverse of  $10^x$  !

• That means, if we say  $f\left(x\right)=10^x$  , then  $f^{-1}\left(x\right)=\log\left(x\right)$  

Multiple Choice

Quick! No calculator!
If 

 $f\left(x\right)=10^x$  
What is f(2)?

• 20

• 200

• 100

• 50

Multiple Choice

Quick! No calculator!

 $f\left(x\right)=10^x$  What is f(3)?

• 300

• 30

• 10000

• 1000

Multiple Choice

 $f^{-1}\left(x\right)=\log\left(x\right)$  
 $f\left(x\right)=10^x$  

What then is log(1000)?

• What? I have no idea.

• 3

• 2

• 10

Concept for your notes

 $y\ =\ 10^x$   and  $\log\left(y\right)=x$   are equivalent statements! Check!
Let x = 2. Then, y will be 100. Check with your calculator: what is log(100)?

Logarithms in words

 $\log\left(1000\right)$  should be thought of as the answer to the question "10 to what power will equal 1000?"
Well, 10 to the third power is 1000, so log(1000) is 3.

Multiple Choice

What is log(10)?

• 1

• 0

• 10

Check

Use your calculator to find what log(5) is.

Check: is 10 to the power of that number equal to 5?

Why logarithms UNDO exponents

 Let look at  $10^x$  .

If we take the logarithm of  $10^x$  , we will get  $\log\left(10^x\right)\ =\ x\log\left(10\right)=x\cdot1=x$  
The logarithm and the exponentiation 'cancel' similar to how if we square root  $x^2$  .

 $\sqrt{x^2}=x$  (this formatting is kinda wack)

Secret info inside of log(x)

By DEFAULT, if you just write log(x), it is the inverse partner function to $10^x$  . 

What if we don't have  $10^x$  , but instead have  $2^x$  ? What is the log partner then?

Notice that $y=2^x$  does not match correctly with the graph of  $y=\log\left(x\right)$  to be its inverse partner. (look closely at the coordinates of a couple points)

The BASE of a logarithm

 $\log x$  is the 'common' logarithm, that partners with an exponential function with a BASE (b-value) of 10.
We really should write  $\log_{10}x$  to show that 10 is the base, but we are lazy.

To be a partner function to a general exponential  $b^x$  , we write a logarithm with a base, like this:  $\log_bx$  

The blue graph is 2^x, the red graph is log₂x and the purple graph is log(x). (also known as log₁₀x)

To make the correct inverse function, we need to do a logarithm with a base that matches the base of the exponential.

Multiple Choice

What is the base of $\log_5\left(2+x\right)$  ?

• 2

• 2+x

• 5

• x

Multiple Choice

Remember when it's missing: What is the base of $\log_{ }x$  ?

• 0

• 10

• x

• 1

Multiple Choice

Convert into words...

 $\log_210$  is equal to: the exponent on ___ that will equal ___

• 10, 2

• 2, 10

Log loop!

If $\log_ba=x$   then  $b^x=a$  

Multiple Choice

If  $\log_43\ =\ x$  , then

•  $3^4=x$  

•  $x^3=4$  

•  $4^x=3$  

•  $3^x=4$  

Multiple Choice

If  $\log_710\ =\ x$  , then

•  $7^{10}=x$  

•  $x^{10}=7$  

•  $10^x=7$  

•  $7^x=10$  

Multiple Choice

If  $\log_4x\ =\ 3$  , then

•  $4^3=x$  

•  $x^4=3$  

•  $4^x=3$  

•  $3^x=4$  

Multiple Choice

What is  $\log_525$  ? (Try writing the equivalent exponent)

• 1/2

• 25

• 5

• 2

 $\log_525\ =\ ?$  
Can be thought of as 
 $5^?=25$  
And you know the answer to that! It must be a power of 2.

Multiple Choice

What is  $\log_2\left(0.5\right)?$  

• 2

• -1

• 0.5

• 0

Multiple Choice

Write the following equation in exponential form:

log₂32 = 5

• 2⁵ = 32

• 2³² = 5

• 5² = 32

• 32² = 5

Multiple Choice

Evaluate: log ₂ 16 = x

• 3

• 2

• 4

• 5

Multiple Choice

Rewrite 10³ = 1000 in logarithmic form.

• log ₁₀₀₀ 3 = 10

• log ₃ 10 = 1000

• log 3 = 1000

• log 1000 = 3

Multiple Choice

Rewrite log₂8 = 3 in exponential form

• 2⁸ = 3

• 2³ = 8

• 3² = 8

• 8³ = 2

Multiple Choice

Rewrite 3⁴ = 81 in logarithmic form.

• log₃4 = 81

• log₈₁3 = 4

• log₃81 = 4

• log₄81 = 3

Multiple Choice

If f^-1(-25) = 7 then what is f(7) =

• 25

• 7

• No enough information.

• -25

Multiple Choice

In an inverse function, the _____ and _______ are switched from the original

• f(x) and g(x)

• input and output

• positives and negatives

• slope and intercept

Multiple Choice

The inverse has been reflected over which line?

• y = x

• y =1

• y = 0

• y = x + 1