Multiplying and dividing Logarithms - Math
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Simplify the expression using logarithmic identities.
Simplify the expression using logarithmic identities.
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The logarithm of a fraction is equal to the logarithm of the numerator minus the logarithm of the denominator.
If we encounter two logarithms with the same base, we can likely combine them. In this case, we can use the reverse of the above identity.
The logarithm of a fraction is equal to the logarithm of the numerator minus the logarithm of the denominator.
If we encounter two logarithms with the same base, we can likely combine them. In this case, we can use the reverse of the above identity.
Which of the following represents a simplified form of 
?
Which of the following represents a simplified form of
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The rule for the addition of logarithms is as follows:
.
As an application of this,
.
The rule for the addition of logarithms is as follows:
As an application of this,
Simplify 
.
Simplify
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Using properties of logs we get:
Using properties of logs we get:
Simplify the following expression:
Simplify the following expression:
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Recall the log rule:
In this particular case, 
and 
. Thus, our answer is 
.
Recall the log rule:
In this particular case,
Use the properties of logarithms to solve the following equation:
Use the properties of logarithms to solve the following equation:
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Since the bases of the logs are the same and the logarithms are added, the arguments can be multiplied together. We then simplify the right side of the equation:
The logarithm can be converted to exponential form:
Factor the equation:
Although there are two solutions to the equation, logarithms cannot be negative. Therefore, the only real solution is 
.
Since the bases of the logs are the same and the logarithms are added, the arguments can be multiplied together. We then simplify the right side of the equation:
The logarithm can be converted to exponential form:
Factor the equation:
Although there are two solutions to the equation, logarithms cannot be negative. Therefore, the only real solution is
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):
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By logarithmic properties:
;
Combining these three terms gives the correct answer:
By logarithmic properties:
Combining these three terms gives the correct answer:
Many textbooks use the following convention for logarithms:
Solve:
Many textbooks use the following convention for logarithms:
Solve:
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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
Find the value of the Logarithmic Expression.
Find the value of the Logarithmic Expression.
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Use the change of base formula to solve this equation.
Use the change of base formula to solve this equation.
Which of the following is equivalent to
?
Which of the following is equivalent to
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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
Our log rules indicate that
So we are really interested in,
Since we are interested in log base
We know that
Therefore, we have
What is another way of expressing the following?
What is another way of expressing the following?
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Use the rule
Use the rule
Expand this logarithm: 
Expand this logarithm:
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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
log of multiple terms is the log of each individual one:
now use the power property to move the exponent over:
Which of the following is equivalent to 
?
Which of the following is equivalent to
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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:
Simplify the expression using logarithmic identities.
Simplify the expression using logarithmic identities.
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The logarithm of a fraction is equal to the logarithm of the numerator minus the logarithm of the denominator.
If we encounter two logarithms with the same base, we can likely combine them. In this case, we can use the reverse of the above identity.
The logarithm of a fraction is equal to the logarithm of the numerator minus the logarithm of the denominator.
If we encounter two logarithms with the same base, we can likely combine them. In this case, we can use the reverse of the above identity.
Which of the following represents a simplified form of ?
Which of the following represents a simplified form of
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The rule for the addition of logarithms is as follows:
.
As an application of this,.
The rule for the addition of logarithms is as follows:
As an application of this,
Simplify .
Simplify
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Using properties of logs we get:
Using properties of logs we get:
Simplify the following expression:
Simplify the following expression:
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Recall the log rule:
In this particular case, and . Thus, our answer is .
Recall the log rule:
In this particular case,
Use the properties of logarithms to solve the following equation:
Use the properties of logarithms to solve the following equation:
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Since the bases of the logs are the same and the logarithms are added, the arguments can be multiplied together. We then simplify the right side of the equation:
The logarithm can be converted to exponential form:
Factor the equation:
Although there are two solutions to the equation, logarithms cannot be negative. Therefore, the only real solution is .
Since the bases of the logs are the same and the logarithms are added, the arguments can be multiplied together. We then simplify the right side of the equation:
The logarithm can be converted to exponential form:
Factor the equation:
Although there are two solutions to the equation, logarithms cannot be negative. Therefore, the only real solution is
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):
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By logarithmic properties:
;
Combining these three terms gives the correct answer:
By logarithmic properties:
Combining these three terms gives the correct answer:
Many textbooks use the following convention for logarithms:
Solve:
Many textbooks use the following convention for logarithms:
Solve:
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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
Find the value of the Logarithmic Expression.
Find the value of the Logarithmic Expression.
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Use the change of base formula to solve this equation.
Use the change of base formula to solve this equation.