### All Precalculus Resources

## Example Questions

### Example Question #1 : Graph Logarithms

What is the domain of the function

**Possible Answers:**

**Correct answer:**

The function is undefined unless . Thus is undefined unless because the function has been shifted left.

### Example Question #11 : Logarithmic Functions

What is the range of the function

**Possible Answers:**

**Correct answer:**

To find the range of this particular function we need to first identify the domain. Since we know that is a bound on our function.

From here we want to find the function value as approaches .

To find this approximate value we will plug in into our original function.

This is our lowest value we will obtain. As we plug in large values we get large function values.

Therefore our range is:

### Example Question #101 : Exponential And Logarithmic Functions

Which of the following logarithmic functions match the provided diagram?

**Possible Answers:**

**Correct answer:**

Looking at the diagram, we can see that when , . Since represents the exponent and represents the product, and any base with an exponent of 1 equals the base, we can determine the base to be 0.5.

### Example Question #1 : Graph Logarithms

Which of the following diagrams represents the graph of the following logarithmic function?

**Possible Answers:**

**Correct answer:**

For , is the exponent of base 5 and is the product. Therefore, when , and when , . As a result, the correct graph will have values of 5 and 125 at and , respectively.

### Example Question #101 : Exponential And Logarithmic Functions

Which of the following diagrams matches the given logarithmic function:

**Possible Answers:**

**Correct answer:**

For this function, represents the exponent and y represents the product of the base 2 and its exponent. On the diagram, it is clear that as the value increases, the value increases exponentially and at , . Those two characteristics of the graph indicate that x is the exponent value and the base is equal to 2.

### Example Question #1 : Graph Logarithms

Which of the following logarithmic functions match the given diagram?

**Possible Answers:**

**Correct answer:**

Looking at the graph, the y-value diminishes exponentially as decreases and increases rapidly as the x-value increases, which indicates that is the exponent value for the equation.

Also, when and when , which can be expressed as and , respectively.

This indicates that the diagram is consistent with the function .

### Example Question #1 : Graph Logarithms

Find .

**Possible Answers:**

**Correct answer:**

Using the Change of Base formula we can rewrite the log as follows.

.

Or,

, then find that .

Therefore,

.

### Example Question #1 : Graph Logarithms

Evaluate the following logarithm:

**Possible Answers:**

**Correct answer:**

The problem asks us to evaluate the following logarithm:

According to the definition of a logarithm, what this equation is asking us is "5 to what power equals 7?":

Using the properties of logarithms, if we take the log of both sides we get the following equation and simplification:

### Example Question #1 : Graph Logarithms

Evaluate the following logarithmic expression:

**Possible Answers:**

**Correct answer:**

In order to evaluate the logarithmic expression, we have to remember the notation of a logarithm and what it means:

Any logarithm expressed in the form above is simply asking "a to what power equals b?" So we're trying to find what power the base must be raised to in order to obtain the value in parentheses. If we look at our expression in particular:

The first term is asking "5 to what power equals 1/25?" while the second is asking "7 to what power equals 49?" Setting these questions up mathematically, we can find the values of each logarithm:

So the value of the first term is -2, and the value of the second term is 2, which gives us:

### Example Question #2 : Graph Logarithms

Evaluate the following logarithm:

**Possible Answers:**

**Correct answer:**

The notation of this logarithm is asking "5 to what power equals 125?" If we set this up mathematically, we can simplify either side until it is readily apparent what the value of x is, which is the value of the logarithm:

So to answer our question, 5 to the power of 3 is 125, so the answer is 3.

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