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MA1310: Week 2 Solving Exponential and Logarithmic Equations

This lab requires you to:

· Use like bases to solve exponential equations.

· Use logarithms to solve exponential equations.

· Use the definition of a logarithm to solve logarithmic equations.

· Use the one-to-one property of logarithms to solve logarithmic equations.

· Solve applied problems involving exponential and logarithmic equations.

· Model exponential growth and decay.

Answer the following questions to complete this lab:

1. Solve the exponential equation by expressing each side as a power of the same base and then equating exponents.

6x = 216

2. Solve the exponential equation. Express the solution in terms of natural logarithms. Then use a calculator to obtain a decimal approximation for the solution.

ex = 22.8

3. Solve the following logarithmic equation. Be sure to reject any value of x that is not in the domain of the original logarithmic expression. Give the exact answer.

log7 x = 2

4. Solve the logarithmic equation. Be sure to reject any value of x that is not in the domain of the original logarithmic expression. Give the exact answer.

log (x + 16) = log x + log 16

5.

Modeling Population: The population of the world has grown rapidly during the past century. As a result, heavy demands have been made on the world's resources. Exponential functions and equations are often used to model this rapid growth, and logarithms are used to model slower growth. The formula models the population of a US state, A, in millions, t years after 2000.

a. What was the population in 2000?

b. When will the population of the state reach 23.3 million?

6.

The goal of our financial security depends on understanding how money in savings accounts grows in remarkable ways as a result of compound interest. Compound interest is computed on your original investment as well as on any accumulated interest. Complete the table for a savings account subject to 4 compounding periods yearly.

Amount Invested

Number of Compounding Periods

Annual Interest Rate

Accumulated Amount

Time t in Years

$15,500

4

5.75%

$30,000

?

7. Cell division is the growth process of many living organisms such as amoebas, plants, and human skin cells. Based on an ideal situation in which no cells die and no by-products are produced, the number of cells present at a given time follows the law of uninhibited growth, which is an Exponential Model.

A0 = A(0): Original Amount (Initial Value)

A colony of bacteria grows according to the law of uninhibited growth. If 100 grams of bacteria are present initially, and 250 grams are present after two hours, how many will be present after 4 hours? Note: Do not round the value of k.

8. Radioactive materials like uranium follow the law of uninhibited decay, which is an Exponential Model. This decay causes radiation. All radioactive substances have a specific half-life, which is the time required for half of the radioactive substance to decay. Uninhibited Radioactive Decay is given by the formula: