Multiplying and Dividing Logarithms - Algebra II
Card 0 of 28
Rewrite the following logarithmic expression in expanded form (i.e. as a sum and/or difference):

Rewrite the following logarithmic expression in expanded form (i.e. as a sum and/or difference):
By logarithmic properties:
;


Combining these three terms gives the correct answer:

By logarithmic properties:
;
Combining these three terms gives the correct answer:
Compare your answer with the correct one above
Many textbooks use the following convention for logarithms:



Solve:

Many textbooks use the following convention for logarithms:
Solve:
Remembering the rules for logarithms, we know that
.
This tells us that
.
This becomes
, which is
.
Remembering the rules for logarithms, we know that .
This tells us that .
This becomes , which is
.
Compare your answer with the correct one above
Find the value of the Logarithmic Expression.

Find the value of the Logarithmic Expression.
Use the change of base formula to solve this equation.





Use the change of base formula to solve this equation.
Compare your answer with the correct one above
Which of the following is equivalent to
?
Which of the following is equivalent to
?
Recall that log implies base
if not indicated.Then, we break up
. Thus, we have
.
Our log rules indicate that
.
So we are really interested in,
.
Since we are interested in log base
, we can solve
without a calculator.
We know that
, and thus by the definition of log we have that
.
Therefore, we have
.
Recall that log implies base if not indicated.Then, we break up
. Thus, we have
.
Our log rules indicate that
.
So we are really interested in,
.
Since we are interested in log base , we can solve
without a calculator.
We know that , and thus by the definition of log we have that
.
Therefore, we have .
Compare your answer with the correct one above
What is another way of expressing the following?

What is another way of expressing the following?
Use the rule


Use the rule
Compare your answer with the correct one above
Expand this logarithm: 
Expand this logarithm:
In order to solve this problem you must understand the product property of logarithms
and the power property of logarithms
. Note that these apply to logs of all bases not just base 10.

log of multiple terms is the log of each individual one:

now use the power property to move the exponent over:

In order to solve this problem you must understand the product property of logarithms and the power property of logarithms
. Note that these apply to logs of all bases not just base 10.
log of multiple terms is the log of each individual one:
now use the power property to move the exponent over:
Compare your answer with the correct one above
Which of the following is equivalent to
?
Which of the following is equivalent to ?
We can rewrite the terms of the inner quantity. Change the negative exponent into a fraction.


This means that:

Split up these logarithms by addition.

According to the log rules, the powers can be transferred in front of the logs as coefficients.
The answer is: 
We can rewrite the terms of the inner quantity. Change the negative exponent into a fraction.
This means that:
Split up these logarithms by addition.
According to the log rules, the powers can be transferred in front of the logs as coefficients.
The answer is:
Compare your answer with the correct one above
Rewrite the following logarithmic expression in expanded form (i.e. as a sum and/or difference):

Rewrite the following logarithmic expression in expanded form (i.e. as a sum and/or difference):
By logarithmic properties:
;


Combining these three terms gives the correct answer:

By logarithmic properties:
;
Combining these three terms gives the correct answer:
Compare your answer with the correct one above
Many textbooks use the following convention for logarithms:



Solve:

Many textbooks use the following convention for logarithms:
Solve:
Remembering the rules for logarithms, we know that
.
This tells us that
.
This becomes
, which is
.
Remembering the rules for logarithms, we know that .
This tells us that .
This becomes , which is
.
Compare your answer with the correct one above
Find the value of the Logarithmic Expression.

Find the value of the Logarithmic Expression.
Use the change of base formula to solve this equation.





Use the change of base formula to solve this equation.
Compare your answer with the correct one above
Which of the following is equivalent to
?
Which of the following is equivalent to
?
Recall that log implies base
if not indicated.Then, we break up
. Thus, we have
.
Our log rules indicate that
.
So we are really interested in,
.
Since we are interested in log base
, we can solve
without a calculator.
We know that
, and thus by the definition of log we have that
.
Therefore, we have
.
Recall that log implies base if not indicated.Then, we break up
. Thus, we have
.
Our log rules indicate that
.
So we are really interested in,
.
Since we are interested in log base , we can solve
without a calculator.
We know that , and thus by the definition of log we have that
.
Therefore, we have .
Compare your answer with the correct one above
What is another way of expressing the following?

What is another way of expressing the following?
Use the rule


Use the rule
Compare your answer with the correct one above
Expand this logarithm: 
Expand this logarithm:
In order to solve this problem you must understand the product property of logarithms
and the power property of logarithms
. Note that these apply to logs of all bases not just base 10.

log of multiple terms is the log of each individual one:

now use the power property to move the exponent over:

In order to solve this problem you must understand the product property of logarithms and the power property of logarithms
. Note that these apply to logs of all bases not just base 10.
log of multiple terms is the log of each individual one:
now use the power property to move the exponent over:
Compare your answer with the correct one above
Which of the following is equivalent to
?
Which of the following is equivalent to ?
We can rewrite the terms of the inner quantity. Change the negative exponent into a fraction.


This means that:

Split up these logarithms by addition.

According to the log rules, the powers can be transferred in front of the logs as coefficients.
The answer is: 
We can rewrite the terms of the inner quantity. Change the negative exponent into a fraction.
This means that:
Split up these logarithms by addition.
According to the log rules, the powers can be transferred in front of the logs as coefficients.
The answer is:
Compare your answer with the correct one above
Rewrite the following logarithmic expression in expanded form (i.e. as a sum and/or difference):

Rewrite the following logarithmic expression in expanded form (i.e. as a sum and/or difference):
By logarithmic properties:
;


Combining these three terms gives the correct answer:

By logarithmic properties:
;
Combining these three terms gives the correct answer:
Compare your answer with the correct one above
Many textbooks use the following convention for logarithms:



Solve:

Many textbooks use the following convention for logarithms:
Solve:
Remembering the rules for logarithms, we know that
.
This tells us that
.
This becomes
, which is
.
Remembering the rules for logarithms, we know that .
This tells us that .
This becomes , which is
.
Compare your answer with the correct one above
Find the value of the Logarithmic Expression.

Find the value of the Logarithmic Expression.
Use the change of base formula to solve this equation.





Use the change of base formula to solve this equation.
Compare your answer with the correct one above
Which of the following is equivalent to
?
Which of the following is equivalent to
?
Recall that log implies base
if not indicated.Then, we break up
. Thus, we have
.
Our log rules indicate that
.
So we are really interested in,
.
Since we are interested in log base
, we can solve
without a calculator.
We know that
, and thus by the definition of log we have that
.
Therefore, we have
.
Recall that log implies base if not indicated.Then, we break up
. Thus, we have
.
Our log rules indicate that
.
So we are really interested in,
.
Since we are interested in log base , we can solve
without a calculator.
We know that , and thus by the definition of log we have that
.
Therefore, we have .
Compare your answer with the correct one above
What is another way of expressing the following?

What is another way of expressing the following?
Use the rule


Use the rule
Compare your answer with the correct one above
Expand this logarithm: 
Expand this logarithm:
In order to solve this problem you must understand the product property of logarithms
and the power property of logarithms
. Note that these apply to logs of all bases not just base 10.

log of multiple terms is the log of each individual one:

now use the power property to move the exponent over:

In order to solve this problem you must understand the product property of logarithms and the power property of logarithms
. Note that these apply to logs of all bases not just base 10.
log of multiple terms is the log of each individual one:
now use the power property to move the exponent over:
Compare your answer with the correct one above