Precalculus : Exponential Equations and Inequalities

Study concepts, example questions & explanations for Precalculus

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Example Questions

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Example Question #1 : Exponential Equations And Inequalities

Solving an exponential equation.

Solve for ,

.

Possible Answers:

Correct answer:

Explanation:

We recall the property:

Now, .

Thus

.

Example Question #1 : Exponential Equations And Inequalities

Solving an exponential equation.

Solve

Possible Answers:

Correct answer:

Explanation:

Use  (which is just , by convention) to solve.

.

Example Question #3 : Exponential Equations And Inequalities

Solve the equation for  using the rules of logarithms.

Possible Answers:

Correct answer:

Explanation:

Expanding the logarithms into sums of logarithms will cancel out the first two x terms, resulting in the equation:

 

Combining the first and second terms, then subtracting the new term over will allow you to isolate the variable term.

Divide both sides of the equation by 2, then exponentiate with 3.

Evaluating this term numerically will give the correct answer.

Example Question #1 : Use Logarithms To Solve Exponential Equations And Inequalities

Solve the following equation:

Possible Answers:

Correct answer:

Explanation:

To solve this equation, recall the following property:

     Can be rewritten as 

Evaluate with your calculator to get

Example Question #5 : Exponential Equations And Inequalities

Solve 

.

Possible Answers:

Correct answer:

Explanation:

After using the division rule to simplify the left hand side you can take the natural log of both sides.

If you then combine like terms you get a quadratic equation which factors to,

 .

Setting each binomial equal to zero and solving for  we get the solution to be .

Example Question #2 : Exponential Equations And Inequalities

If you invest  into a savings account which earns an interest rate  per year, how much would it take for your deposit to double?

Possible Answers:

 

 

Correct answer:

 

Explanation:

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

Example Question #2 : Compound Interest Problems

If you deposit  into a savings account which earns a  yearly interest rate, how much is in your account after two years?

Possible Answers:

Correct answer:

Explanation:

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:

Example Question #1 : Compound Interest Problems

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?

Possible Answers:

Correct answer:

Explanation:

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:

Example Question #4 : Compound Interest Problems

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)

Possible Answers:

Correct answer:

Explanation:

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 .

Example Question #5 : Compound Interest Problems

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?

 

Possible Answers:

None of the other answers.

Correct answer:

Explanation:

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  

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