### All Precalculus Resources

## Example Questions

### Example Question #1 : Solve Exponential Equations

Solve an equation involving exponents and logarithms.

Solve for .

**Possible Answers:**

**Correct answer:**

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 #1 : Exponential Functions

Solve an exponential equation.

Solve for .

**Possible Answers:**

**Correct answer:**

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 #2 : Solve Exponential Equations

Solve for .

**Possible Answers:**

**Correct answer:**

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:

.

### Example Question #4 : Solve Exponential Equations

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.

**Possible Answers:**

2010: fish

2015: fish

2010: fish

2015: fish

2010: fish

2015: fish

2010: fish

2015: fish

**Correct answer:**

2010: fish

2015: fish

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 #1 : Solve Exponential Equations

Solve for using properties of exponents.

**Possible Answers:**

**Correct answer:**

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:

**Possible Answers:**

None of the other answers.

**Correct answer:**

Combine the constants:

Isolate the exponential function by dividing:

Take the natural log of both sides:

Finally isolate x:

### Example Question #7 : Solve Exponential Equations

Solve the equation for .

**Possible Answers:**

**Correct answer:**

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

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