Exponential and Logarithmic Functions - Pre-Calculus
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The amount of phosphorus present in a sample at a given time is given by the following equation:
Where 
is in days, 
is the amount of phosphorus after 
days, and 
is the initial amount of phosphorus at the beginning of the first day. What percent of the initial amount of phosphorus is left after 
days of decay?
The amount of phosphorus present in a sample at a given time is given by the following equation:
Where
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The problem asks us for the percent that the amount of phosphorus after t days is of the original amount of phosphorus. If we think of this percentage with respect to the variables present in the equation, we can see that the following fraction expresses the amount of phosphorus after t days as a percentage of the initial amount of phosphorus:
So if this is the fraction we want to solve for, we should divide both sides of the equation by 
to obtain this fraction on the left side of the equation:
We now have the fraction we want to solve for in terms of just one variable, 
, for which we plug in 25 days to find the percentage of phosphorus left of the initial amount after 25 days:
So after 25 days of decay, the amount of phosphorus is 47% of the initial amount.
The problem asks us for the percent that the amount of phosphorus after t days is of the original amount of phosphorus. If we think of this percentage with respect to the variables present in the equation, we can see that the following fraction expresses the amount of phosphorus after t days as a percentage of the initial amount of phosphorus:
So if this is the fraction we want to solve for, we should divide both sides of the equation by
We now have the fraction we want to solve for in terms of just one variable,
So after 25 days of decay, the amount of phosphorus is 47% of the initial amount.
The exponential decay of an element is given by the following function:
Where 
is the amount of the element left after 
days, and 
is the initial amount of the element. If there are 
kg of the element left after 
days, what was the initial amount of the element?
The exponential decay of an element is given by the following function:
Where
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The problem asks us for the initial amount of the element, so first let's solve our equation for 
:
The problem tells us that 25 days has passed, which gives us 
, and it also tells us the amount left after 25 days, which gives us 
. Now that we have our equation for 
, we can plug in the given values to find the initial amount of the element:
kg
The problem asks us for the initial amount of the element, so first let's solve our equation for
The problem tells us that 25 days has passed, which gives us
The exponential decay of an element is given by the function:
In this function, 
is the amount of the element left after 
days, and 
is the initial amount of the element. If 
of the element is left after seven days, how much of the element was there to begin with, rounded to the nearest kilogram?
The exponential decay of an element is given by the function:
In this function,
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To find the initial amount, you must rearrange the equation to solve for 
:
Divide both sides by 
:
Substituting in the values from the problem gives
To find the initial amount, you must rearrange the equation to solve for
Divide both sides by
Substituting in the values from the problem gives
The exponential decay of an element is given by the function
In this function, 
is the amount left after 
days, and 
is the initial amount of the element. What percent of the element is left after ten days, rounded to the nearest whole percent?
The exponential decay of an element is given by the function
In this function,
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To find the final percentage of the element left, we must rearrange the equation to solve for 
:
Now, using the ten days as 
, we can solve for the percent of the element left after ten days:
To find the final percentage of the element left, we must rearrange the equation to solve for
Now, using the ten days as
The exponential decay of an element is given by the function
where 
is the amount of the element after 
days, and 
is the initial amount of the element. If 
of the element are left after four days, how much of the element was there initially, to the nearest tenth of a kilogram?
The exponential decay of an element is given by the function
where
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To solve for the initial amount, we must use rearrange the equation:
We now substitute the values given from the problem
To solve for the initial amount, we must use rearrange the equation:
We now substitute the values given from the problem
If you invest 
into a savings account which earns an interest rate 
per year, how much would it take for your deposit to double?
If you invest
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The equation of the value for this problem is
.
We can divide by R to get
.
We want to solve for n in this case, which is the amount of years. If we use the natural log on both sides and properties of logarithms, we get
.
If we solve for n, we get
The equation of the value for this problem is
We can divide by R to get
We want to solve for n in this case, which is the amount of years. If we use the natural log on both sides and properties of logarithms, we get
If we solve for n, we get
If you deposit 
into a savings account which earns a 
yearly interest rate, how much is in your account after two years?
If you deposit
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Since we are investing for two years with a yearly rate of 5%, we will use the formula to calculate compound interest.
where
is the amount of money after time.
is the principal amount (initial amount).
is the interest rate.
is time.
Our amount after two years is:
Since we are investing for two years with a yearly rate of 5%, we will use the formula to calculate compound interest.
where
Our amount after two years is:
If you deposit 
into a savings account which compounds interest every month, what is the expression for the amount of money in your account after 
years if you earn a nominal interest rate of 
compounded monthly?
If you deposit
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Since 
is the nominal interest rate compounded monthly we write the interest term as 
as it is the effective monthly rate.
We compound for 
years which is 
months. Since our interest rate is compounded monthly our time needs to be in the same units thus, months will be the units of time.
Plugging this into the equation for compound interest gives us the expression:
Since
We compound for
Plugging this into the equation for compound interest gives us the expression:
John opens a savings account and deposits 
into it. This savings account gains 
interest per year. After 
years, John withdraws all the money, and deposits it into another savings account with 
interest per year. 
years later, John withdraws the money.
How much money does John have after this 
year period? (Assume compound interest in both accounts)
John opens a savings account and deposits
How much money does John have after this
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Plugging our numbers into the formula for compound interest, we have:
.
So John has about 
after the first three years.
After placing his money into the other savings account, he has
after 
more years.
So John has accumulated about 
.
Plugging our numbers into the formula for compound interest, we have:
So John has about
After placing his money into the other savings account, he has
So John has accumulated about
How many years does it take for 
to grow into 
when the 
is deposited into a savings account that gains 
annual compound interest?
How many years does it take for
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The correct answer is about 
years.
To find the number of years required, we solve the compound interest formula for 
.
The formula is as follows:
Substitute known values.
Divide by 
.
Take the natural log of both sides.
Use the log power rule.
Divide by 
.
use a calculator to simplify.
The correct answer is about
To find the number of years required, we solve the compound interest formula for
The formula is as follows:
Substitute known values.
Divide by
Take the natural log of both sides.
Use the log power rule.
Divide by
Suppose you took out a loan 
years ago that gains 
interest. Suppose that you haven't made any payments on it yet, and right now you owe 
on the loan. How much was the loan worth when you took it out?
Suppose you took out a loan
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The formula for the compund interest is as follows:
By substuting known values into the compound interest formula, we have:
.
From here, substitute known values.
Divide by 
The formula for the compund interest is as follows:
By substuting known values into the compound interest formula, we have:
From here, substitute known values.
Divide by
There were 240mg of caffeine in the discontinued energy drink. The decay rate for caffeine in the human bloodstream is around 0.14. If Jackie drinks this energy drink around 8PM, how much caffeine will still be in her system at midnight?
There were 240mg of caffeine in the discontinued energy drink. The decay rate for caffeine in the human bloodstream is around 0.14. If Jackie drinks this energy drink around 8PM, how much caffeine will still be in her system at midnight?
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Because this is a process taking place in the human body, we should use the exponential decay formula involving e:
where A is the current amount, P is the initial amount, r is the rate of growth/decay, and t is time.
In this case, since the amount of caffeine is decreasing rather than increasing, use 
. Between 8PM and midnight, 4 hours pass, so use 
. The initial amount of caffeine is given as 240 mg, so use 
.
Now evaluate:
Because this is a process taking place in the human body, we should use the exponential decay formula involving e:
where A is the current amount, P is the initial amount, r is the rate of growth/decay, and t is time.
In this case, since the amount of caffeine is decreasing rather than increasing, use
Now evaluate:
Stuff animals were a strange craze of the 90's. A Cat stuff animal with white paws sold for $6 in 1997. In 2015, the Cat will sell for $1015. What has been the approximate rate of growth for these stuff animal felines?
Stuff animals were a strange craze of the 90's. A Cat stuff animal with white paws sold for $6 in 1997. In 2015, the Cat will sell for $1015. What has been the approximate rate of growth for these stuff animal felines?
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Use the formula for exponential growth 
where y is the current value, A is the initial value, r is the rate of growth, and t is time. Between 1997 and 2015, 18 years passed, so use 
. The stuffed animal was originally worth $6, so 
. It is now worth $1,015, so 
.
Our equation is now:
divide by 6:
take both sides to the power of 
:
subtract 1
As a percent, r is about 33%.
Use the formula for exponential growth
Our equation is now:
As a percent, r is about 33%.
Simplify the function:
Simplify the function:
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When an exponent is raised to the power of another exponent, just multiply the exponents together.
When an exponent is raised to the power of another exponent, just multiply the exponents together.
Simplify the function:
Simplify the function:
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When an exponent is raised to the power of another exponent, just multiply the exponents together.
When an exponent is raised to the power of another exponent, just multiply the exponents together.
Simplify
Simplify
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By the property of the addition of logarithms with the same base
As such
By the property of the addition of logarithms with the same base
As such
Simplify
Simplify
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.
Simplify the expression:
.
Simplify the expression:
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First, you can begin to simplfy the numerator by converting all 3 expressions into base 2.
, which simplifies to
For the denominator, the same method applies. Convert the 25 into base 5, and when simplified becomes simply 5.
The final simplified answer becomes:
First, you can begin to simplfy the numerator by converting all 3 expressions into base 2.
For the denominator, the same method applies. Convert the 25 into base 5, and when simplified becomes simply 5.
The final simplified answer becomes:
Evaluate the following expression using knowledge of the properties of exponents:
Evaluate the following expression using knowledge of the properties of exponents:
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Let's work through this equation involving exponents one term at a time. The first term we see is 
, for which we can apply the following property:
So if we plug our values into the formula for the property, we get:
Because 
. Our next term is 
, for which we'll need the property:
Using the values for our term, we have:
The third term of the equation is 
, for which the quickest way to evaluate would be using the following property:
Using the values from our term, this gives us:
The next property we will need to consider for our fourth term is given below:
If we plug in the corresponding values from our term, we get:
Finally, our last term requires knowledge of the following simple property: Any number raised to the power of zero is 1. With this in mind, our last term becomes:
Rewriting the equation with all of the values we've just evaluated, we obtain our final answer:
Let's work through this equation involving exponents one term at a time. The first term we see is
So if we plug our values into the formula for the property, we get:
Because
Using the values for our term, we have:
The third term of the equation is
Using the values from our term, this gives us:
The next property we will need to consider for our fourth term is given below:
If we plug in the corresponding values from our term, we get:
Finally, our last term requires knowledge of the following simple property: Any number raised to the power of zero is 1. With this in mind, our last term becomes:
Rewriting the equation with all of the values we've just evaluated, we obtain our final answer:
Simplify the expression.
Simplify the expression.
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Using the properties of exponents, we can either choose to subtract the exponents of the corresponding bases or rewrite the expression using negative exponents as such:
Here, we combine the terms with corresponding bases by adding the exponents together to get
Placing the x term (since it has a negative exponent) in the denominator will result in the correct answer. It can be shown that simply subtracting the exponents of corresponding bases will result in the same answer.
Using the properties of exponents, we can either choose to subtract the exponents of the corresponding bases or rewrite the expression using negative exponents as such:
Here, we combine the terms with corresponding bases by adding the exponents together to get
Placing the x term (since it has a negative exponent) in the denominator will result in the correct answer. It can be shown that simply subtracting the exponents of corresponding bases will result in the same answer.