Exponentials and LogsModules 12, 13, 14, 15

October 23, 2012

Logs and exponentials are inverses of each other and can be rewritten in this way:

We can use the opposite function to isolate our variable when we solve equations.

Inverse Functions

Exponential functions are of the form:

Our variable here is still x. Ex.

What is an exponential function?

where a > 0 and a ≠ 1.

Log functions are of the form:

“What power do I raise the base a to in order to get the argument x?”

Ex.

What is a log function?

where a > 0 and a ≠ 1.

Exponential and log functions can also have transformations just like the functions did from the first exam material.

Ex.

Transformations

Is this a valid exponential function?

𝑓 (𝑥 )=(−3)𝑥

No, because here the base value would have to be a = -3, and we know that a has to be positive.

Is this a valid exponential function?

𝑓 (𝑥 )=(−3)𝑥

Is this a valid exponential function?

𝑓 (𝑥 )=−3𝑥

Yes, because our base is a=3, which is valid. The negative out front is a reflection over the x-axis because it’s not being raised to the x power.

Is this a valid exponential function?

𝑓 (𝑥 )=−3𝑥

Is this a valid exponential function?

𝑓 (𝑥 )=( 23 )𝑥

Yes, because our base is a=2/3, which is valid because fractions are okay.

Is this a valid exponential function?

𝑓 (𝑥 )=( 23 )𝑥

Is this a valid log function?

𝑓 (𝑥 )=𝑙𝑜𝑔𝜋 (2𝑥 )

Yes, because our base is a=π, which is valid because it is a value positive and not equal to 1. The 2 in front of the x is a horizontal transformation, which causes the graph to compress horizontally by ½.

Is this a valid log function?

𝑓 (𝑥 )=𝑙𝑜𝑔𝜋 (2𝑥 )

Is this a valid log function?

𝑓 (𝑥 )=𝑙𝑜𝑔7(𝑥4)

Yes, because our base is a=7, which is valid because it is a value positive and not equal to 1. The 4 can come out front by log rules, and so it will end up vertically stretching by 4.

Is this a valid log function?

𝑓 (𝑥 )=𝑙𝑜𝑔7(𝑥4)

Is this a valid log function?

𝑓 (𝑥 )=𝑙𝑜𝑔−2(𝑥 )

No, because our base is a = -2, and negative numbers aren’t allowed.

Is this a valid log function?

𝑓 (𝑥 )=𝑙𝑜𝑔−2(𝑥 )

Log Properties: Identity:

Inverse (I): ◦

Inverse (II):

Exponent to Constant:

Product:

Quotient:

Both types of function act differently when a is between 0 and 1 than they do when a is above 1.

Think about what happens when we square a number:

gets smaller

gets bigger

So you need to memorize 4 basic graph shapes.

Graphing

𝑓 (𝑥 )=𝑎𝑥 ,0<𝑎<1 𝑓 (𝑥 )=𝑎𝑥 ,𝑎>1

𝑓 (𝑥 )=log𝑎 𝑥 ,0<𝑎<1 𝑓 (𝑥 )=log𝑎 𝑥 ,𝑎>1

Graphing Notes Also think about the asymptotes to help you

think about where they end up when you transform the graphs.

AND… Our good old friends domain and range.

𝑓 (𝑥 )=𝑎𝑥 ,0<𝑎<1 𝑓 (𝑥 )=𝑎𝑥 ,𝑎>1

𝑓 (𝑥 )=log𝑎 𝑥 ,0<𝑎<1 𝑓 (𝑥 )=log𝑎 𝑥 ,𝑎>1

𝑓 (𝑥 )=𝑎𝑥 ,0<𝑎<1 𝑓 (𝑥 )=𝑎𝑥 ,𝑎>1

𝑓 (𝑥 )=log𝑎 𝑥 ,0<𝑎<1 𝑓 (𝑥 )=log𝑎 𝑥 ,𝑎>1

Domain: all realsRange: (0, infinity)

Domain: all realsRange: (0, infinity)

Domain: (0, infinity)Range: all reals

Domain: (0, infinity)Range: all reals

Graph a function:

◦ First we have a base of a=2, so that tells us we need to start with the graph of . Then we’ll reflect it over the x axis to turn it into , before we shift the whole thing up by 3 units to get to our final answer of .

Graph a function:

◦ First we have a base of a=2, so that tells us we need to start with the graph of . Then we’ll reflect it over the x axis to turn it into , before we shift the whole thing up by 3 units to get to our final answer of .

New asymptote will be horizontal and at y=3 because it started at y=0 and there was a 3 unit vertical change.

Domain: all realsRange: (- infinity, 3)

Remember that logs and exponentials are inverse functions and we can use them to “undo” each other. Sometimes we need to do this, sometimes we don’t. There are usually a couple ways to solve.

For example:

Solving Equations

realize that 4 = 2^2

when we have the same base, we can set the arguments equal to each other

Solving Equations

Solving Equations When we have variables in the exponent, we need to take the log of both sides to “get it out” so we can solve for it.

Log rules

Solving Equations We want to get x by itself, so we need to raise both sides to the x power (or reorganize using the definition).

Take the cube root of both sides.

Sometimes we have really complicated logs which we can expand into many individual terms using log rules.

- -

Expanding Logs

We also want to be able to pull many logs into a single log in other situations.

Note that we cannot combine them unless they have the same base!

+ +

Condensing Logs

Sometimes we’re in the situation when we need an exact value of a log using our scientific calculator, but the base is something besides e or 10.

We can rewrite any log using this formula so that it’s possible to compute:

◦ Ex. = 0.7924 = 2.3219

Rewriting Bases