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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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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 means it is not the root of some number).

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 which belong to functions. i.e. (x,y) becomes (y,x)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 which belong to functions. i.e. (x,y) becomes (y,x)

How to graph the logarithmic

function.


f(x)=2 which belong to functions. i.e. (x,y) becomes (y,x)x

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 2 which belong to functions. i.e. (x,y) becomes (y,x)x

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 2 which belong to functions. i.e. (x,y) becomes (y,x)x

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 which belong to functions. i.e. (x,y) becomes (y,x)

Changing from exponential

form to logarithmic form.


Both of these are referred to as bases which belong to functions. i.e. (x,y) becomes (y,x)

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


log which belong to functions. i.e. (x,y) becomes (y,x)2(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 which belong to functions. i.e. (x,y) becomes (y,x)

Four log properties.


There are a few truths about logs which we will call which belong to functions. i.e. (x,y) becomes (y,x)

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 which belong to functions. i.e. (x,y) becomes (y,x)

1. Solve for x: log3x = log3 4

X = 34

X = 4

Click on the green arrow of the correct answer above.


No, x = 3 which belong to functions. i.e. (x,y) becomes (y,x)4 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! which belong to functions. i.e. (x,y) becomes (y,x)

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 which belong to functions. i.e. (x,y) becomes (y,x)

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. which belong to functions. i.e. (x,y) becomes (y,x)

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! which belong to functions. i.e. (x,y) becomes (y,x)

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 which belong to functions. i.e. (x,y) becomes (y,x)

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. which belong to functions. i.e. (x,y) becomes (y,x)

Using properties 1,2 and 3:

ln 1 = 0

log10 = 1

log2 2-4 = -4

which totals to -3

Go back.


Way to go! which belong to functions. i.e. (x,y) becomes (y,x)

Using properties 1,2 and 3:

ln 1 = 0

log10 = 1

log2 2-4 = -4

which totals to -3


Concept 6 which belong to functions. i.e. (x,y) becomes (y,x)

The three expansion properties of logs.


The 3 expansion properties of logs which belong to functions. i.e. (x,y) becomes (y,x)

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 which belong to functions. i.e. (x,y) becomes (y,x)

1. loga(uv) = logau + logav

2.

3.


Example 1 which belong to functions. i.e. (x,y) becomes (y,x)

Expand to single log expressions:

Applying property 1

Log 10 = 1 from the 2nd property which we

had earlier.


Example 2 which belong to functions. i.e. (x,y) becomes (y,x)

Expand to single log expressions:

Applying property 2

Applying property 1 and from before ln e = 1


Example 3 which belong to functions. i.e. (x,y) becomes (y,x)

Expand to single log expressions:

First change the radical to an exponent.

Next, apply property 3


Next, apply property 2 for quotients which belong to functions. i.e. (x,y) becomes (y,x)


Next, apply property 3to the exponentials. which belong to functions. i.e. (x,y) becomes (y,x)

Now they

are single

logs.


Your turn: which belong to functions. i.e. (x,y) becomes (y,x)

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 which belong to functions. i.e. (x,y) becomes (y,x)

previous slide.


The first step is to use property 1 which which belong to functions. i.e. (x,y) becomes (y,x)

will expand to:

Now we use property 3

This is the final answer.


No, this is not the final answer. which belong to functions. i.e. (x,y) becomes (y,x)

Return to the previous slide and

click on the correct answer.


Yes, we now use property 3 to expand which belong to functions. i.e. (x,y) becomes (y,x)

further to:

This is the final answer.

This is still not the final answer.


Nope, we are not done yet. which belong to functions. i.e. (x,y) becomes (y,x)

Return


Whenever the base of the log matches the which belong to functions. i.e. (x,y) becomes (y,x)

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


is the property. which belong to functions. i.e. (x,y) becomes (y,x)

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: an expanded log and condense it back to a single log expression.

Always begin by reversing property 3

Next use property 2

which is a single log


Condense to a single log: an expanded log and condense it back to a single log expression.

reversing property 3

Next use property 2

which is a single log


Concept 7 an expanded log and condense it back to a single log expression.

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 log may find the value of the 25

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: may find the value of the

We are given base a, and we

change to base c.


Example 1 may find the value of the

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 may find the value of the

calculator, but not all.

1. Find

using the property

2. Find

using the same

property


Some problems can be done without a may find the value of the

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! may find the value of the

Find ln 10


Right! may find the value of the

while

as found using a calculator.

Which of the following is false

concerning

None are false


Uses the change of base formula may find the value of the

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


Concept 8 may find the value of the

Solving exponential equations with logs


Solve may find the value of the

Take the ln of both sides.

Property: lnex = x

Solve

Isolate the exponential expression.

Take the log base 10 of both sides.


Solve may find the value of the

Isolate the exponential expression.

Take the ln of both sides

Property: lnex = x


Solve may find the value of the

Isolate the exponential expression.

Take the log base 10 of both sides.


Concept 9 may find the value of the

Solving logarithm equations with exponents


Solve may find the value of the

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 may find the value of the

Apply the expansion property for exponents.

Isolate the ln expression.

Change to the exponential form

Solve for x


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