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By Dr. Julia Arnold

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Review of Logs

By Dr. Julia Arnold

Concept 1

The Exponential Function

x 2x

0 1

1 2

2 4

-1 1/2

-2 1/4

The exponential function f with base a is denoted by f(x) = ax where a > 0 a 1, and x is any real number.

To graph a specific exponential function we will use a table of values.

This is the graph of 2x

Pi or is what is called a transcendental number ( which means it is not the root of some number).

is another such number. On the graphing calculator, you can find e by pushing the yellow 2nd function button and the Ln key. On the display you will see e^( Type 1 and close parenthesis. Thus

e^( 1). Press Enter and you will see 2.718281828 which represents an approximation of e.

ex is called the natural exponential function.

Concept 2

The exponential function is the inverse of the logarithm function

Recall that Inverse Functions reverse the ordered pairs which belong to functions. i.e. (x,y) becomes (y,x)

The log function is the inverse of the exponential function.

If the exponential is 2x, then its inverse is log2 (x) (read log x base 2 ).

its inverse is log3(x)

If the exponential is 3x...

Read log x base 3

If the exponential is 10x...

its inverse is logx

Read log x

Base 10 is considered the common base and thus log x is

the common log and as such the base is omitted.

If the exponential is 2x=y, then its inverse is log2 (x)=y (read log y base 2 =‘s x ).

its inverse is log3(x)

If the exponential is 3x...

its inverse is logx

If the exponential is 10x...

Base 10 is considered the common base and thus log y is

the common log and as such the base is omitted.

If the exponential is ex...

its inverse is lnx

Read l n x

Base e is considered the natural base and thus ln x is

the natural log and is written ln to distinguish it from log.

Concept 3

How to graph the logarithmic

function.

f(x)=2x

Let’s look at the two graphs of the exponential and the

logarithmic function:

Goes through (0,1) which means 1 = 20

The domain is all real numbers.

The range is all positive real numbers.

Goes through (1,0) which means 0 = log2(1)

The domain is all positive real numbers.

The range is all real numbers.

f(x)=log2(x)

x 2x

0 1

1 2

2 4

-1 1/2

-2 1/4

f(x)=2x

Remember this slide?

We used a table of values to graph the exponential y = 2x.

Since we know that the y = log2(x) is the inverse of the function above, we can just switch the ordered pairs in the table above and create the log graph for base 2.

x 2x

0 1

1 2

2 4

-1 1/2

-2 1/4

f(x)=2x

x

log2(x)

y = log2(x) is the inverse of the function y = 2x. To create the graph we can just switch the ordered pairs in the table left and create the log graph for base 2.

Switch

This is the easy way to do a log graph.

1 0

2 1

4 2

1/2 -1

1/4 -2

Concept 4

Changing from exponential

form to logarithmic form.

Both of these are referred to as bases

Y is the exponent

on the left.

Logs are = to the exponent

on the right.

First task is to be able to go from exponential form to

logarithmic form.

x = ay becomes y = loga(x)

log2(32)=5

Thus, 25 = 32 becomes

Read: log 32 base 2 =‘s 5

log2(32) = 5

Thus, 25 = 32 becomes

log3(81) = 4

34 = 81 becomes

log2(1/8) = -3

2-3 = 1/8 becomes

log5(1/25) = -2

5-2 = 1/25 becomes

log2(1) = 0

20 = 1 becomes

log(1000) = 3

103 = 1000 becomes

e1 = e becomes

ln (e) = 1

Concept 5

Four log properties.

There are a few truths about logs which we will call

properties:

1. loga(1) = 0 for any a > 0 and not equal to 1

because a0=1 (exponential form of log form)

2. loga(a) = 1 for any a > 0 and not equal to 1

because a1=a (exponential form of log form)

3. loga(ax) = x for any a > 0 and not equal to 1

because ax=ax (exponential form of log form)

4. If loga x = loga y , then x = y.

Practice Problems

1. Solve for x: log3x = log3 4

X = 34

X = 4

Click on the green arrow of the correct answer above.

No, x = 34 is not correct.

Use the 4th property:

4. If loga x = loga y , then x = y.

log3x = log3 4, then x = 4

Go back.

Way to go!

Using the 4th property:

4. If loga x = loga y , then x = y

you concluded correctly that

x = 4 for log3x = log3 4.

Practice Problems

2. Solve for x: log21/8 = x

X = -3

X = 3

Click on the green arrow of the correct answer above.

No, x = 3 is not correct.

Use the 3rd property:

3. loga(ax) = x

log21/8 = log2 8-1 = log2 (23)-1 =

log2 2-3 then x = -3 since the 2’s make a match.

Go back.

Way to go!

Using the 3rd property:

3. loga(ax) = x

log21/8 = log2 8-1 = log2 (23)-1 =

log2 2-3 then x = -3 since the 2’s make a match.

Practice Problems

3. Evaluate: ln 1 + log 10 - log2(24)

-2

-3

Click on the green arrow of the correct answer above.

No, -2 is not correct.

Using properties 1,2 and 3:

ln 1 = 0

log10 = 1

log2 2-4 = -4

which totals to -3

Go back.

Way to go!

Using properties 1,2 and 3:

ln 1 = 0

log10 = 1

log2 2-4 = -4

which totals to -3

Concept 6

The three expansion properties of logs.

The 3 expansion properties of logs

1. loga(uv) = logau + logav

Proof: Set logau =x and logav =y then change to exponential form.

ax = u and ay = v.

ax+y =ax ay = uv so, write

ax+y = uv in log form

loga(uv) = x + y

but that’s logau =x and logav =y , so write

loga(uv) = logau + logav

which is the result we were looking for.

Do you see how this property relates to the

exponential property?

The 3 expansion properties of logs

1. loga(uv) = logau + logav

2.

3.

Example 1

Expand to single log expressions:

Applying property 1

Log 10 = 1 from the 2nd property which we

had earlier.

Example 2

Expand to single log expressions:

Applying property 2

Applying property 1 and from before ln e = 1

Example 3

Expand to single log expressions:

First change the radical to an exponent.

Next, apply property 3

Next, apply property 2 for quotients

Next, apply property 3to the exponentials.

Now they

are single

logs.

Your turn:

Expand to single logs:

The first step is to use property 1

The first step is to use property 3

No, incorrect, return to the

previous slide.

The first step is to use property 1 which

will expand to:

Now we use property 3

This is the final answer.

No, this is not the final answer.

Return to the previous slide and

click on the correct answer.

Yes, we now use property 3 to expand

further to:

This is the final answer.

This is still not the final answer.

Nope, we are not done yet.

Return

Whenever the base of the log matches the

number you are taking the log of, the answer is the exponent on the number which is 1 in this case.

is the property.

Or from the beginning of the problem you

could have said:

We can use the same 3 expansion properties of logs to take an expanded log and condense it back to a single log expression.

1. logau + logav =loga(uv)

2.

3.

Condense to a single log:

Always begin by reversing property 3

Next use property 2

which is a single log

Condense to a single log:

reversing property 3

Next use property 2

which is a single log

Concept 7

Finding logs on your calculator for

any base number.

On your graphing calculator or scientific calculator, you may find the value of the

log (of a number) to the base 10 or the

ln(of a number) to the base e by simply pressing

the appropriate button.

What if you want to find the value of a log to a different base?

How can we find for example, the log25

How can we find for example, the log25

Set log25 = x

Change to exponential form 2x = 5

Take the log (base 10 ) of both sides.

log 2x = log 5

x log 2 = log 5 using the 3rd expansion property

thus x =

This shows us how we can create the change of base

formula:

Changes the base to c.

The change of base formula:

We are given base a, and we

change to base c.

Example 1

Find the following value using

a calculator:

Since the calculator is built to

find base 10 or base e, choose either one and use the change of base formula.

Some problems can be done without a

calculator, but not all.

1. Find

using the property

2. Find

using the same

property

Some problems can be done without a

calculator, but not all.

1. Find

Click on the correct slide to advance to the next slide. Click on the wrong side and you will remain here.

Right!

Find ln 10

Right!

while

as found using a calculator.

Which of the following is false

concerning

None are false

Uses the change of base formula

Because of the log properties all of the statements below are true

Concept 8

Solving exponential equations with logs

Solve

Take the ln of both sides.

Property: lnex = x

Solve

Isolate the exponential expression.

Take the log base 10 of both sides.

Solve

Isolate the exponential expression.

Take the ln of both sides

Property: lnex = x

Solve

Isolate the exponential expression.

Take the log base 10 of both sides.

Concept 9

Solving logarithm equations with exponents

Solve

Remember property 4. If loga x = loga y , then x = y.

That property applies in this problem, thus

x - 1 = 3

x = 4

Solve

Change to exponential form

Solve

Apply the expansion property for exponents.

Isolate the ln expression.

Change to the exponential form

Solve for x

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