Solving and Graphing Exponential Equations - Algebra 2

Although the bases are not the same, we know that . Therefore we now have

Now, we can add the exponents.

When we add exponents, we try to factor to see if we can simplify it. Let's factor . We get . Remember to apply the rule of multiplying exponents which is to add the exponents and keeping the base the same. Next, we can divide on both sides.

We know

With the same base, we know that .

The first step in solving an equation like this to make the base the same on both sides of the equation. Since 5 is a factor of 125, we can rewrite the equation like this:

Using the Power of a Power Property of exponents, we get:

If the bases are the same on both sides of the equation, then the exponents must be equal, so

becomes

Solving for x:

In order to solve for the unknown variable, first change the base of the value of 25 to .

The equation becomes:

Since the bases are now the same, we can set the powers equal to each other.

Simplify the right side by distributing the integer through the binomial.

Subtract from both sides.

Add twelve on both sides.

Divide by fifteen on both sides and reduce the fraction.

The answer is:

Rewrite the right side of the equation using a base of ten.

One thousand to the power of x can be rewritten using the product of exponents.

Now that the bases are equal, set the powers equal to each other.

Subtract from both sides.

Simplify both sides.

The answer is:

In order to solve this equation, we will need to convert the nine on the right side to a base of three.

Rewrite the equation.

Set the powers equal to each other since the bases are common.

Distribute the two across the binomial.

Subtract from both sides.

The answer is:

In order to determine the value of x, we will need to convert the base of one-tenth to the base of ten.

Rewrite the fraction as a base of ten.

Rewrite the right side of the equation using the new term.

According to the product rule for exponents, we can set the powers equal since the bases are similar.

Divide by negative one on both sides.

Subtract on both sides. The equation will become:

Divide by negative two on both sides.

The answer is:

In order to solve this, we will need to rewrite the right side as the similar base to the left side of the equation.

Rewrite the right side.

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

Subtract on both sides.

Add six on both sides.

Divide by two on both sides.

The answer is:

To solve this equation, we will need to convert the 100 into base ten.

Rewrite the number using this base.

Now that the bases are similar, the exponents can be set equal to each other.

Simplify this equation.

Subtract on both sides.

The answer is:

To be able to set the powers equal to each other, we will need common bases.

Convert eight into two cubed.

Set the powers equal to each other.

Divide by three on both sides.

Subtract from both sides.

Divide by five on both sides.

The answer is:

Rewrite the right side with a negative exponent. The goal is to establish similar bases to set the powers equal to each other.

Set the powers equal to each other.

Subtract one from both sides.

Divide by three sides.

The answer is:

To solve this equation, I would first rewrite 8 as a base of 2:

Now, plug back into the equation and simplify. When there are two exponents next to each other like this, multiply them:

Since the bases are the same, you can set the exponents equal to each other:

In order to solve this, the bases of the powers will need to be converted. Notice that both terms can be rewritten as base three.

Rewrite the equation.

Now that the bases are equal to each other, the powers can be set equal to each other.

Divide negative one on both sides. This will move the negative to the other side.

Subtract on both sides.

Divide by negative 18 on both sides.

The answer is:

The asymptote of this equation can be found by observing that regardless of . We are thus solving for the value of as approaches zero.

So the value that cannot exceed is , and the line is the asymptote.

An exponential equation of the form has only one asymptote - a horizontal one at . In the given function, , so its one and only asymptote is .

To find the vertical asymptotes, we set the denominator of the function equal to zero and solve.

For positive values, increases exponentially in the direction and goes to positive infinity, so there is no asymptote on the positive -axis. For negative values, as decreases, the term becomes closer and closer to zero so approaches as we move along the negative axis. As the graph below shows, this is forms a horizontal asymptote.

Factorize both the numerator and denominator.

Notice that one of the binomials will cancel.

The domain of this equation cannot include .

The simplified equation is:

Since the term canceled, the term will have a hole instead of an asymptote.

Set the denominator equal to zero.

Subtract one from both sides.

There will be an asymptote at only:

The answer is:

Factor the numerator and denominator.

Notice that the terms will cancel. The hole will be located at because this is a removable discontinuity.

The denominator cannot be equal to zero. Set the denominator to find the location where the x-variable cannot exist.

The asymptote is located at .