Understanding Logarithms - Algebra 2
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Based on the definition of logarithms, what is 
?
Based on the definition of logarithms, what is
For any equation 
, 
. Thus, we are trying to determine what power of 10 is 1000. 
, so our answer is 3.
For any equation
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Solve for 
in the equation:
Solve for
This question tests your understanding of log functions.
can be converted to the form 
.
In this problem, make sure to divide both sides by 
in order to put it in the above form, where 
. Remember 
.
Therefore,
This question tests your understanding of log functions.
In this problem, make sure to divide both sides by
Therefore,
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Which of the following expressions is equivalent to the expression 
?
Which of the following expressions is equivalent to the expression
By the reverse-FOIL method, we factor the polynomial as follows:
Therefore, we can use the property
as follows:
By the reverse-FOIL method, we factor the polynomial as follows:
Therefore, we can use the property
as follows:
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Evaluate 
.
Evaluate
Take the common logarithm of both sides, and take advantage of the property of the logarithm of a power:
Take the common logarithm of both sides, and take advantage of the property of the logarithm of a power:
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Evaluate 
.
Evaluate
Take the common logarithm of both sides, and take advantage of the property of the logarithm of a power:
Take the common logarithm of both sides, and take advantage of the property of the logarithm of a power:
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Many textbooks use the following convention for logarithms:
What is the value of 
?
Many textbooks use the following convention for logarithms:
What is the value of
Remember:
is the same as saying 
.
So when we ask "What is the value of 
?", all we're asking is "10 raised to which power equals 1,000?" Or, in an expression:
.
From this, it should be easy to see that 
.
Remember:
So when we ask "What is the value of
From this, it should be easy to see that
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What is the value of 
?
Round to the nearest hundreth.
What is the value of
Round to the nearest hundreth.
Base-10 logarithms are very easy if the operands are a power of 
. Begin by rewriting the question:
Becomes...
because 
Applying logarithm rules, you can factor out the 
:
Now, 
is 
.
Therefore, your answer is 
.
Base-10 logarithms are very easy if the operands are a power of
Becomes...
because
Applying logarithm rules, you can factor out the
Now,
Therefore, your answer is
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What is the value of 
?
What is the value of
Base-10 logarithms are very easy if the operands are a power of 
. Begin by rewriting the question:
Becomes...
because 
Applying logarithm rules, you can factor out the 
:
Now, 
is 
.
Therefore, your answer is 
.
Base-10 logarithms are very easy if the operands are a power of
Becomes...
because
Applying logarithm rules, you can factor out the
Now,
Therefore, your answer is
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Evaluate the following expression:
Evaluate the following expression:
Without a subscript a logarithmic expression is base 10.
The expression 
The logarithmic expression is asking 10 raised to what power equals 1000 or what is x when
We know that 
so 
Without a subscript a logarithmic expression is base 10.
The expression
The logarithmic expression is asking 10 raised to what power equals 1000 or what is x when
We know that
so
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Assuming the value of 
is positive, simplify: 
Assuming the value of
Rewrite the logarithm in division.
As a log property, we can pull down the exponent of the power in front as the coefficient.
Cancel out the 
.
The answer is: 
Rewrite the logarithm in division.
As a log property, we can pull down the exponent of the power in front as the coefficient.
Cancel out the
The answer is:
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Solve the following:
Solve the following:
When the base isn't explicitly defined, the log is base 10. For our problem, the first term
is asking:
For the second term,
is asking:
So, our final answer is 
When the base isn't explicitly defined, the log is base 10. For our problem, the first term
is asking:
For the second term,
is asking:
So, our final answer is
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Evaluate 
.
Evaluate
The first thing we can do is bring the exponent out of the log, to the front:
Next, we evaluate 
:
Recall that log without a specified base is base 10 thus
.
Therefore
becomes,
.
Finally, we do the simple multiplication:
The first thing we can do is bring the exponent out of the log, to the front:
Next, we evaluate
Recall that log without a specified base is base 10 thus
Therefore
becomes,
Finally, we do the simple multiplication:
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Simplify 
Simplify
One of the properties of logs is the ability to cancel out terms based on the base of the log. Since the base of the log is 10 we can simplify the 100 to 10 squared.
The log base 10 and the 10 cancel out, leaving you with the value of the exponent, 2 as the answer.
One of the properties of logs is the ability to cancel out terms based on the base of the log. Since the base of the log is 10 we can simplify the 100 to 10 squared.
The log base 10 and the 10 cancel out, leaving you with the value of the exponent, 2 as the answer.
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Evaluate: 
Evaluate:
In order to eliminate the log based ten, we will need to raise both sides as the exponents using the base of ten.
The equation becomes:
The ten and log based ten will cancel, leaving just the power on the left side. Change the negative exponent into a fraction on the right side.
Divide by two on both sides, which is similar to multiplying by a half on both sides.
Simplify both sides.
The answer is: 
In order to eliminate the log based ten, we will need to raise both sides as the exponents using the base of ten.
The equation becomes:
The ten and log based ten will cancel, leaving just the power on the left side. Change the negative exponent into a fraction on the right side.
Divide by two on both sides, which is similar to multiplying by a half on both sides.
Simplify both sides.
The answer is:
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Solve: 
Solve:
Break up 
using log rules. The log has a default base of ten.
The exponent can be brought down as the coefficient since the bases of the second term are common.
This means that: 
The answer is: 
Break up
The exponent can be brought down as the coefficient since the bases of the second term are common.
This means that:
The answer is:
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What is the value of 
?
What is the value of
In order to solve this, we will need to rewrite the base 
as 
, since log is by default base 10.
Rewrite the expression.
By log rules, the exponent can be pulled down as the coefficient.
The answer is: 
In order to solve this, we will need to rewrite the base
Rewrite the expression.
By log rules, the exponent can be pulled down as the coefficient.
The answer is:
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Evaluate: 
Evaluate:
The log term has a default base of 10. The 1000 will need to be rewritten as base 10.
Raise the coefficient of the log term as the power.
According to the log property:
The log based 10 and the 10 inside the quantity of the log will cancel, leaving just the power.
The answer is: 
The log term has a default base of 10. The 1000 will need to be rewritten as base 10.
Raise the coefficient of the log term as the power.
According to the log property:
The log based 10 and the 10 inside the quantity of the log will cancel, leaving just the power.
The answer is:
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Simplify: 
Simplify:
The log has a default of base ten. This means we should convert the 1000 to a common base 10.
Replace this value inside the log term.
Since the log base 10 and the ten to a certain power are existent, they will both cancel, leaving just the power itself.
The answer is: 
The log has a default of base ten. This means we should convert the 1000 to a common base 10.
Replace this value inside the log term.
Since the log base 10 and the ten to a certain power are existent, they will both cancel, leaving just the power itself.
The answer is:
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Solve: 
Solve:
Change the base of the inner term or log to base ten.
According to the log property:
The log based ten and the ten to the power of will cancel, leaving just the power.
The answer is: 
Change the base of the inner term or log to base ten.
According to the log property:
The log based ten and the ten to the power of will cancel, leaving just the power.
The answer is:
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Evaluate: 
Evaluate:
Rewrite the log such that it is in its simplest form. Break up the 500 with common factors.
This can be broken into addition of logs.
The answer is: 
Rewrite the log such that it is in its simplest form. Break up the 500 with common factors.
This can be broken into addition of logs.
The answer is:
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