We begin with the following:

This can be rewritten as

Recall that if you have two exponents with equal bases, you can simply set the exponents equal to eachother. Do so to get the following:

Solve this to get t.

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.

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.

The first step is to make sure we don't have a zero on one side which we can easily take care of:

Now we can take the logarithm of both sides using natural log:

Note: we can apply the Power Rule here

Step 1: Rewrite the right side as a power of :

Step 2: Rewrite the original equation:

Step 3: Since the bases are equal, I can set the exponents equal.

So,

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:

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 .