Solving Exponential Equations - Algebra II

When dealing with exponential equations, we want to make sure the bases are the same. This way we can set-up an equation with the exponents.

With the same base, we can now write

Subtract on both sides.

Divide on both sides.

Because both sides of the equation have the same base, set the terms equal to each other.

Add 9 to both sides:

Then, subtract 2x from both sides:

Finally, divide both sides by 3:

Begin by recognizing that both sides of the equation have a root term of .

Using the power rule, we can set the exponents equal to each other.

Begin by recognizing that both sides of the equation have the same root term, .

We can use the power rule to combine exponents.

Set the exponents equal to each other.

Solve for the values of a and b:

In 2009, and (zero years since 2009). Plug this into the exponential equation form:

. Solve for to get .

In 2013, and . Therefore,

or . Solve for to get

.

Then the exponential growth function is

.

8 and 4 are both powers of 2.

125 and 25 are both powers of 5.

Therefore, the equation can be rewritten as

.

Using the Distributive Property,

.

Since both sides now have the same base, set the two exponents equal to one another and solve:

Add 30 to both sides:

Add to both sides:

Divide both sides by 20:

Both 27 and 9 are powers of 3, therefore the equation can be rewritten as

.

Using the Distributive Property,

.

Now that both sides have the same base, set the two exponenents equal and solve.

Add 12 to both sides:

Subtract from both sides:

The first step in thist problem is divide both sides by three: . Then, recognize that 8 could be rewritten with a base of 2 as well (). Therefore, your answer is 3.

Let's convert to base .

We know the following:

Simplify.

Solve.

Let's convert to base .

We know the following:

Simplify.

Solve.

.

When multiplying exponents with the same base, we will apply the power rule of exponents:

We will simply add the exponents and keep the base the same.

When multiplying exponents with the same base, we will apply the power rule of exponents:

We will simply add the exponents and keep the base the same.

Simplify.

Solve.

When adding exponents with the same base, we need to see if we can factor out the numbers of the base.

In this case, let's factor out .

We get the following:

Since we are now multiplying with the same base, we get the following expression:

Now we have the same base and we just focus on the exponents.

The equation is now:

Solve.

First, we need to convert to base .

We know .

Therefore we can write the following expression:

.

Next, when we add exponents of the same base, we need to see if we can factor out terms.

In this case, let's factor out .

We get the following:

.

Since we are now multiplying with the same base, we get the following expression:

.

Now we have the same base and we just focus on the exponents.

The equation is now:

Solve.

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.

With the same base, we can rewrite as .

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.

With the same base, we can rewrite as .

Add on both sides.

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.

With the same base, we can rewrite as .

Add on both sides.

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.

With the same base, we can rewrite as .

When multiplying exponents with the same base, we add the exponents and keep the base the same.

We can just rewrite as such: