Solve Exponential Equations - Pre-Calculus

First, use the additive property of exponents to simplify the right side of the equation.

.

Thus,

.

Now, take the natural log of both sides

.

Use the multiplicative property of logarithms to expand the left side to get

Now, apply the logarithms to the exponents

.

Rearrange to get the x-terms on one side

.

Finally, divide the 2 on both sides

.

First, let's begin by simplifying the left hand side.

becomes and becomes . Remember that , and the in that expression can come out to the front, as in .

Now, our expression is

From this, we can cancel out the 2's and an x from both sides.

Thus our answer becomes:

.

In 2010, in our equation because we have had no years past 2010. Plugging that in to the model equation and solving:

, since anything raised to the power of zero becomes . So the population of fish in 2010 is fish.

In 2015, because 5 years have passed since 2010. Plugging that into our equation and solving gives us

So the population of fish in 2015 is fish. This is an example of exponential decay since the function is decreasing.

Since , the equation simplifies to .

Since the bases are equal, we can then set the exponents equal to each other.

Solving for x in this simple equation gives the correct answer.

Combine the constants:

Isolate the exponential function by dividing:

Take the natural log of both sides:

Finally isolate x:

The key to this is that . From here, the equation can be factored as if it were .

and

and

and

Now take the natural log (ln) of the two equations.

and

and

First, simplify the left side of the equation using the additive rule for exponents.

.

Our equation now becomes:

Equating we set the exponents equal to eachother and solve.

Thus,

First, use the additive property of exponents to simplify the right side of the equation.

.

Thus,

.

Now, take the natural log of both sides

.

Use the multiplicative property of logarithms to expand the left side to get

Now, apply the logarithms to the exponents

.

Rearrange to get the x-terms on one side

.

Finally, divide the 2 on both sides

.

First, let's begin by simplifying the left hand side.

becomes and becomes . Remember that , and the in that expression can come out to the front, as in .

Now, our expression is

From this, we can cancel out the 2's and an x from both sides.

Thus our answer becomes:

.

In 2010, in our equation because we have had no years past 2010. Plugging that in to the model equation and solving:

, since anything raised to the power of zero becomes . So the population of fish in 2010 is fish.

In 2015, because 5 years have passed since 2010. Plugging that into our equation and solving gives us

So the population of fish in 2015 is fish. This is an example of exponential decay since the function is decreasing.

Since , the equation simplifies to .

Since the bases are equal, we can then set the exponents equal to each other.

Solving for x in this simple equation gives the correct answer.

Combine the constants:

Isolate the exponential function by dividing:

Take the natural log of both sides:

Finally isolate x:

The key to this is that . From here, the equation can be factored as if it were .

and

and

and

Now take the natural log (ln) of the two equations.

and

and

First, simplify the left side of the equation using the additive rule for exponents.

.

Our equation now becomes:

Equating we set the exponents equal to eachother and solve.

Thus,

First, use the additive property of exponents to simplify the right side of the equation.

.

Thus,

.

Now, take the natural log of both sides

.

Use the multiplicative property of logarithms to expand the left side to get

Now, apply the logarithms to the exponents

.

Rearrange to get the x-terms on one side

.

Finally, divide the 2 on both sides

.

First, let's begin by simplifying the left hand side.

becomes and becomes . Remember that , and the in that expression can come out to the front, as in .

Now, our expression is

From this, we can cancel out the 2's and an x from both sides.

Thus our answer becomes:

.

In 2010, in our equation because we have had no years past 2010. Plugging that in to the model equation and solving:

, since anything raised to the power of zero becomes . So the population of fish in 2010 is fish.

In 2015, because 5 years have passed since 2010. Plugging that into our equation and solving gives us

So the population of fish in 2015 is fish. This is an example of exponential decay since the function is decreasing.

Since , the equation simplifies to .

Since the bases are equal, we can then set the exponents equal to each other.

Solving for x in this simple equation gives the correct answer.

Combine the constants:

Isolate the exponential function by dividing:

Take the natural log of both sides:

Finally isolate x: