Review of Logs
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Review of Logs. By Dr. Julia Arnold. Concept 1 The Exponential Function. x 2 x 0 1 1 2 2 4 -1 1/2 -2 1/4. The exponential function f with base a is denoted by f(x) = a x where a > 0 a 1, and x is any real number.

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By Dr. Julia Arnold

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By dr julia arnold

Review of Logs

By Dr. Julia Arnold


By dr julia arnold

Concept 1

The Exponential Function


By dr julia arnold

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


By dr julia arnold

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.


By dr julia arnold

Concept 2

The exponential function is the inverse of the logarithm function


By dr julia arnold

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.


By dr julia arnold

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.


By dr julia arnold

Concept 3

How to graph the logarithmic

function.


By dr julia arnold

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)


By dr julia arnold

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.


By dr julia arnold

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


By dr julia arnold

Concept 4

Changing from exponential

form to logarithmic form.


By dr julia arnold

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


By dr julia arnold

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


By dr julia arnold

Concept 5

Four log properties.


By dr julia arnold

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.


By dr julia arnold

Practice Problems

1. Solve for x: log3x = log3 4

X = 34

X = 4

Click on the green arrow of the correct answer above.


By dr julia arnold

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.


By dr julia arnold

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.


By dr julia arnold

Practice Problems

2. Solve for x: log21/8 = x

X = -3

X = 3

Click on the green arrow of the correct answer above.


By dr julia arnold

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.


By dr julia arnold

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.


By dr julia arnold

Practice Problems

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

-2

-3

Click on the green arrow of the correct answer above.


By dr julia arnold

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.


By dr julia arnold

Way to go!

Using properties 1,2 and 3:

ln 1 = 0

log10 = 1

log2 2-4 = -4

which totals to -3


By dr julia arnold

Concept 6

The three expansion properties of logs.


By dr julia arnold

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?


By dr julia arnold

The 3 expansion properties of logs

1. loga(uv) = logau + logav

2.

3.


By dr julia arnold

Example 1

Expand to single log expressions:

Applying property 1

Log 10 = 1 from the 2nd property which we

had earlier.


By dr julia arnold

Example 2

Expand to single log expressions:

Applying property 2

Applying property 1 and from before ln e = 1


By dr julia arnold

Example 3

Expand to single log expressions:

First change the radical to an exponent.

Next, apply property 3


By dr julia arnold

Next, apply property 2 for quotients


By dr julia arnold

Next, apply property 3to the exponentials.

Now they

are single

logs.


By dr julia arnold

Your turn:

Expand to single logs:

The first step is to use property 1

The first step is to use property 3


By dr julia arnold

No, incorrect, return to the

previous slide.


By dr julia arnold

The first step is to use property 1 which

will expand to:

Now we use property 3

This is the final answer.


By dr julia arnold

No, this is not the final answer.

Return to the previous slide and

click on the correct answer.


By dr julia arnold

Yes, we now use property 3 to expand

further to:

This is the final answer.

This is still not the final answer.


By dr julia arnold

Nope, we are not done yet.

Return


By dr julia arnold

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.


By dr julia arnold

is the property.

Or from the beginning of the problem you

could have said:


By dr julia arnold

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.


By dr julia arnold

Condense to a single log:

Always begin by reversing property 3

Next use property 2

which is a single log


By dr julia arnold

Condense to a single log:

reversing property 3

Next use property 2

which is a single log


By dr julia arnold

Concept 7

Finding logs on your calculator for

any base number.


By dr julia arnold

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


By dr julia arnold

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.


By dr julia arnold

The change of base formula:

We are given base a, and we

change to base c.


By dr julia arnold

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.


By dr julia arnold

Some problems can be done without a

calculator, but not all.

1. Find

using the property

2. Find

using the same

property


By dr julia arnold

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.


By dr julia arnold

Right!

Find ln 10


By dr julia arnold

Right!

while

as found using a calculator.

Which of the following is false

concerning

None are false


By dr julia arnold

Uses the change of base formula

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


By dr julia arnold

Concept 8

Solving exponential equations with logs


By dr julia arnold

Solve

Take the ln of both sides.

Property: lnex = x

Solve

Isolate the exponential expression.

Take the log base 10 of both sides.


By dr julia arnold

Solve

Isolate the exponential expression.

Take the ln of both sides

Property: lnex = x


By dr julia arnold

Solve

Isolate the exponential expression.

Take the log base 10 of both sides.


By dr julia arnold

Concept 9

Solving logarithm equations with exponents


By dr julia arnold

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


By dr julia arnold

Solve

Apply the expansion property for exponents.

Isolate the ln expression.

Change to the exponential form

Solve for x


By dr julia arnold

Please send any comments about this power point to [email protected]

Thanks for viewing.


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