Natural Log - Algebra 2
Card 1 of 88
Solve for 
.
Solve for
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The first thing we notice about this problem is that 
is an exponent. This should be an immediate reminder: use logs!
The question is, which base should we choose for the log? We should use the natural log (log base e) because the right-hand side of the equation already has e as a base of an exponent. As you will see, things cancel out more nicely this way.
Take the natural log of both sides:
Rewrite the right-hand side of the equation using the product rule for logs:
Now rewrite the whole equation after bringing down those exponents.
is the same thing as 
, which equals 1.
Now we just divide by 
on both sides to isolate 
.
The first thing we notice about this problem is that
The question is, which base should we choose for the log? We should use the natural log (log base e) because the right-hand side of the equation already has e as a base of an exponent. As you will see, things cancel out more nicely this way.
Take the natural log of both sides:
Rewrite the right-hand side of the equation using the product rule for logs:
Now rewrite the whole equation after bringing down those exponents.
Now we just divide by
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Rewrite as a single logarithmic expression:
Rewrite as a single logarithmic expression:
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Using the properties of logarithms
and 
,
we simplify as follows:
Using the properties of logarithms
we simplify as follows:
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Which of the following expressions is equal to the expression 
?
Which of the following expressions is equal to the expression
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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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What are the domain and the range of the function 
?
What are the domain and the range of the function
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Remember that 
is still a logarithm of a positive number, 
.
It's not possible to raise 
to ANY power and obtain a negative number. Because even 
, for example, is just 
, which is a ratio of two positive numbers, and therefore positive.
More than that, it's also not possible to obtain 0 by raising 
to any power. Think: "To what power can I exponentiate e and obtain 0?"
So the domain is strictly positive. It excludes negative numbers and 0.
What about the range? To what possible values are we allowed to exponentiate 
?
Well, we just saw that 
has a definition for negative numbers. 
(this fact is true for ALL numbers, not just 
).
And we can obviously raise it to positive powers. So the range is all real numbers. It includes negative numbers, 0, and positive numbers.
Remember that
It's not possible to raise
More than that, it's also not possible to obtain 0 by raising
So the domain is strictly positive. It excludes negative numbers and 0.
What about the range? To what possible values are we allowed to exponentiate
Well, we just saw that
And we can obviously raise it to positive powers. So the range is all real numbers. It includes negative numbers, 0, and positive numbers.
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Solve 
. Round to the nearest thousandth.
Solve
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The original equation is:
Subtract 
from both sides:
Divde both sides by 
:
Take the natural logarithm of both sides:
Divde both sides by 
and use a calculator to get:
The original equation is:
Subtract
Divde both sides by
Take the natural logarithm of both sides:
Divde both sides by
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Solve for 
:
.
If necessary, round to the nearest tenth.
Solve for
If necessary, round to the nearest tenth.
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Give both sides the same base, using e:
.
Because e and ln cancel each other out, 
.
Solve for x and round to the nearest tenth:
Give both sides the same base, using e:
Because e and ln cancel each other out,
Solve for x and round to the nearest tenth:
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Solve for x:
Solve for x:
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To solve for x, keep in mind that the natural logarithm and the exponential cancel each other out (property of any logarithm with a base that is being taken of that same base with an exponent attached). When they cancel, we are just left with the exponents:
To solve for x, keep in mind that the natural logarithm and the exponential cancel each other out (property of any logarithm with a base that is being taken of that same base with an exponent attached). When they cancel, we are just left with the exponents:
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Determine the value of: 
Determine the value of:
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The natural log has a base of 
. This means that the term will simplify to whatever is the power of 
. Some examples are:
This means that 
Multiply this quantity with three.
The answer is: 
The natural log has a base of
This means that
Multiply this quantity with three.
The answer is:
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Determine the value of: 
Determine the value of:
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In order to simplify this expression, use the following natural log rule.
The natural log has a default base of 
. This means that:
The answer is: 
In order to simplify this expression, use the following natural log rule.
The natural log has a default base of
The answer is:
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Simplify: 
Simplify:
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According to log properties, the coefficient 
in front of the natural log can be rewritten as the exponent raised by the quantity inside the log.
Notice that natural log has a base of 
. This means that raising the log by base 
will eliminate both the 
and the natural log.
The terms become: 
Simplify the power.
The answer is: 
According to log properties, the coefficient
Notice that natural log has a base of
The terms become:
Simplify the power.
The answer is:
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Evaluate: 
Evaluate:
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The natural log has a default base of 
. Natural log to of an exponential raised to the power will be just the power. The natural log and 
will be eliminated.
Rewrite the expression.
The answer is: 
The natural log has a default base of
Rewrite the expression.
The answer is:
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Determine the value of: 
Determine the value of:
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The natural log has a default base of 
.
According to the rule of logs, we can use:
The coefficient in front of the natural log can be transferred as the power of the exponent.
The natural log and base e will cancel, leaving just the exponent.
The answer is: 
The natural log has a default base of
According to the rule of logs, we can use:
The coefficient in front of the natural log can be transferred as the power of the exponent.
The natural log and base e will cancel, leaving just the exponent.
The answer is:
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Solve: 
Solve:
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The natural log has a default base of 
. This means that the natural log of 
to the certain power will be just the power itself.
The expression 
becomes:
The answer is: 
The natural log has a default base of
The expression
The answer is:
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Simplify: 
Simplify:
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Use the log properties to separate each term. When the terms inside are multiplied, the logs can be added.
Rewrite the expression.
The exponent, 7 can be dropped as the coefficient in front of the natural log. Natural log of the exponential is equal to one since the natural log has a default base of 
.
The answer is: 
Use the log properties to separate each term. When the terms inside are multiplied, the logs can be added.
Rewrite the expression.
The exponent, 7 can be dropped as the coefficient in front of the natural log. Natural log of the exponential is equal to one since the natural log has a default base of
The answer is:
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The equation 
represents Newton's Law of Cooling. Solve for 
.
The equation
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Use base 
and raise both the left and right sides as the powers of 
. This will eliminate the natural log term.
The equation becomes:
Multiply the quantity 
on both sides.
Add 
on both sides.
The equation becomes:
To isolate 
, divide 
on both sides.
The answer is: 
Use base
The equation becomes:
Multiply the quantity
Add
The equation becomes:
To isolate
The answer is:
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Evaluate: 
Evaluate:
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The natural log has a default base of 
.
Use the log property:
We can cancel the base and the log of the base.
The expression 
becomes:
The answer is: 
The natural log has a default base of
Use the log property:
We can cancel the base and the log of the base.
The expression
The answer is:
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Simplify: 
Simplify:
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Notice that the terms inside the log are added, with a common factor of 
. Pull out a common factor.
Notice that these two terms inside the log are multiplied. We can split the log into two terms.
The value of the first term is equal to one, since natural log has a default base of 
. We can use the property 
to eliminate the log and the 
term, which will cancel leaving just the power of one.
The expression becomes: 
We cannot use the property of logs to simplify the second term.
The answer is: 
Notice that the terms inside the log are added, with a common factor of
Notice that these two terms inside the log are multiplied. We can split the log into two terms.
The value of the first term is equal to one, since natural log has a default base of
The expression becomes:
We cannot use the property of logs to simplify the second term.
The answer is:
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Solve the expression: 
Solve the expression:
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In order to eliminate the natural log and solve for x, we will need to exponential both sides because 
is the base of natural log.
The left side will be reduced to just the inner quantity of the natural log.
Subtract 
from both sides of the equation.
Divide by two on both sides.
The answer is: 
In order to eliminate the natural log and solve for x, we will need to exponential both sides because
The left side will be reduced to just the inner quantity of the natural log.
Subtract
Divide by two on both sides.
The answer is:
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Evaluate: 
Evaluate:
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The natural log has a default base of 
.
This means that: 
According to the property of logs, 
, and will equal just the power.
Simplify the expression.
The answer is: 
The natural log has a default base of
This means that:
According to the property of logs,
Simplify the expression.
The answer is:
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Evaluate: 
Evaluate:
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Simplify the first term. The natural log has a default base of 
.
According to log rules:
This means that: 
The answer is: 
Simplify the first term. The natural log has a default base of
According to log rules:
This means that:
The answer is:
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