Algebra II : Solving and Graphing Exponential Equations

Example Questions

Example Question #104 : Solving Exponential Equations

Solve:

Explanation:

In order to solve the equation, we will need to change the bases of both sides.

Rewrite the equation using these two bases.

Now that both sides have equal bases, we can set the exponents alike.

Simplify both sides.

Subtract nine from both sides, and then divide both sides by nine.

Example Question #105 : Solving Exponential Equations

Solve:

Explanation:

Change the base of the fraction to base 2.

Now that both bases are similar, we can set the powers equal to each other.

Use distribution to simplify the right side.

Divide by negative nine on both sides.

Reduce this fraction.

Example Question #106 : Solving Exponential Equations

Evaluate:

Explanation:

Change the base of the second number to base two.

We can then replace the term and use the power rule of exponents to simplify the equation.

Set the powers equal now that we have same bases.

Subtract 6 from both sides.

Divide by eight on both sides.

Reduce the fractions.

Example Question #107 : Solving Exponential Equations

Evaluate:

Explanation:

Convert the bases to  to some power on both sides.

Rewrite the terms of the equation.

With similar bases, we can set the powers equal to each other.

Simplify the right side by distribution.

Subtract six on both sides, and divide by negative six.

Example Question #108 : Solving Exponential Equations

Solve:

Explanation:

We can rewrite the fractional exponent as a radical.

Raise both sides by the power of four to eliminate the radical.

Divide by two on both sides.

Reduce both fractions.

Example Question #109 : Solving Exponential Equations

Solve:

Explanation:

To be able to solve this equation, we will need to change the bases on both sides to a common base.

Choose three as the common base.

Replace the terms into the equation.

With the common bases, we can set the powers equal to each other.

Distribute the four on the right side.

Subtract 20 on both sides.

Divide by negative 12 on both sides.

Reduce both fractions.

Example Question #110 : Solving Exponential Equations

Solve:

Explanation:

Convert the bases to a common base so that the exponential powers can be set equal to each other.  We can choose base two and rewrite both sides using exponential powers.

The radical is to the power of one-half.

Set the powers equal to each other.

Distribute the outer terms into the binomials.

Multiply by two on both sides to eliminate the fraction.

Subtract 40 on both sides.

Divide by 6 on both sides.

Example Question #1 : Graphing Exponential Functions

Give the -intercept of the graph of the equation .

The graph has no -intercept.

The graph has no -intercept.

Explanation:

Set  and solve for

We need not work further. It is impossible to raise a positive number 2 to any real power to obtain a negative number. Therefore, the equation has no solution, and the graph of  has no -intercept.

Example Question #2 : Graphing Exponential Functions

What is/are the asymptote(s) of the graph of the function ?

and

Explanation:

An exponential function of the form

has as its one and only asymptote the horizontal line

Since we define  as

,

then

and the only asymptote is the line of the equation .

Example Question #3 : Graphing Exponential Functions

Determine whether each function represents exponential decay or growth.

a) growth

b) growth

a) decay

b) decay

a) growth

b) decay

a) decay

b) growth

a) decay

b) growth

Explanation:

a)

This is exponential decay since the base, , is between  and .

b)

This is exponential growth since the base, , is greater than .