# Precalculus : Solve Exponential Equations

## Example Questions

### Example Question #6 : Exponential Functions

Solve an equation involving exponents and logarithms.

Solve for .

Explanation:

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,

### Example Question #2 : Exponential Functions

Solve an exponential equation.

Solve for .

Explanation:

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

.

### Example Question #121 : Exponential And Logarithmic Functions

Solve for

Explanation:

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.

.

### Example Question #9 : Exponential Functions

The population of fish in a pond is modeled by the exponential function

, where  is the population of fish and  is the number of years since January 2010.

Determine the population of fish in January 2010 and January 2015.

2010:  fish

2015:  fish

2010:  fish

2015:  fish

2010:  fish

2015:  fish

2010:  fish

2015:  fish

2010:  fish

2015:  fish

Explanation:

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.

### Example Question #10 : Exponential Functions

Solve for  using properties of exponents.

Explanation:

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.

### Example Question #1 : Solve Exponential Equations

Solve:

Explanation:

Combine the constants:

Isolate the exponential function by dividing:

Take the natural log of both sides:

Finally isolate x:

### Example Question #2 : Solve Exponential Equations

Solve the equation for .

Explanation:

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