MA131
0
: Module
2
Exponential a nd Logarithmic Functions
Exercise 2
.2
Solving Exponential and Logarithmic
Equations
1
Answer the following questions to complete this exercise:
1.
Solve the following exponential equation by expressing each side as a power of the same base and then equating exponents:
6
x
= 216
2.
Solve the following exponential equation: e x
= 22.8
Express the solution in terms of natural logarithms. Then, use a calculator to obtain a decimal approximation for the solution.
3.
Solve the following logarithmic equation: log 7 x = 2
Reject any value of x that is not in the domain of the original logarithmic expression. Give the exact answer. 4.
Solve the following logarithmic equation: log ( x + 16) = log x + log 16
Reject any value of x that is not in the domain of the original logarithmic expression.
Give the exact answer. 5.
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
0.0547
16.6 t Ae

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 four compounding periods yearly.
Use the following formula to solve this problem:
1
nt r AP n 







MA131
0
: Module
2
Exponential a nd Logarithmic Functions
Exercise 2
.2
Solving Exponential and Logarithmic
Equations
2
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 in 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 created, the number of cells present at a given time follows the law of uninhibited growth, which is an exponential model.
00
( ) or ; 0 ktkt ftAeAAek
  
A
0
=
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 materia ls 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 the radioactive substance to decay. Uninhibited radio active decay is given by the formula:
0
( ) , 0 kt AtAek

The half
-
life of thorium
-
229 is 7,340 years. How long will it take for a sample of this substance to decay to 20 percent of its original amount?