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 .

a)

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

b)

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

a)

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

b)

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

When solving exponential equations, we need to ensure that we have the same base. When that happens, our equations our based on the exponents.

With same base, we can write:

Subtract on both sides.

Divide on both sides.

To find the -intercept(s) of , we need to set the numerator equal to zero and solve.

First, notice that can be factored into . Now set that equal to zero: .

Since we have two sets in parentheses, there are two separate values that can cause our equation to equal zero: one where and one where .

Solve for each value:

and

.

Therefore there are two -interecpts: and .

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.

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 solving exponential equations, we need to ensure that we have the same base. When that happens, our equations our based on the exponents.

With the same base, we can now write

Add and subtract on both sides.

When solving exponential equations, we need to ensure that we have the same base. When that happens, our equations our based on the exponents.

With same base, we can write:

Subtract on both sides.

Divide on both sides.

The question gives us P (48,000) and asks us to find t (time). We can substitute for P and start to solve for t:

Now we have to isolate t by taking the natural log of both sides:

And since , t can easily be isolated:

Note: does not equal . You have to perform the log operation first before dividing.