# Exponential and Logarithmic Functions

## Logarithmic Properties


### Learning Objectives

1. Simplify expressions using the properties for exponents. (IA 5.2.1)
2. Use the properties of logarithms. (IA 10.4.1)

### Objective 1: Simplify expressions using the properties for exponents (IA 5.2.1)


### The Product Property

Simplify expressions using the properties for exponents.
To multiply powers with the same base we need to ________ exponents.

This leads us to the Product Property  

### The Quotient Property

Simplify  
To divide powers with the same base we need to __________ exponents.
This leads us to the Quotient Property  

### The Power Property

Simplify  
To raise a power to a power we need to __________ exponents.

This leads us to the Power Property  .
We will also use these other properties:


### Practice Makes Perfect

Simplify expressions using the properties for exponents.

### Objective 2: Use the properties of logarithms (IA 10.4.1).


### Practice Makes Perfect

In chemistry, pH is used as a measure of the acidity or alkalinity of a substance. The pH scale runs from 0 to 14. Substances with a pH less than 7 are considered acidic, and substances with a pH greater than 7 are said to be basic. Our bodies, for instance, must maintain a pH close to 7.35 in order for enzymes to work properly. To get a feel for what is acidic and what is basic, consider the following pH levels of some common substances:
1. Battery acid: 0.8
2. Stomach acid: 2.7
3. Orange juice: 3.3
4. Pure water: 7 (at 25° C)
5. Human blood: 7.35
6. Fresh coconut: 7.8
7. Sodium hydroxide (lye): 14
To determine whether a solution is acidic or basic, we find its pH, which is a measure of the number of active positive hydrogen ions in the solution. The pH is defined by the following formula, where 
 is the concentration of hydrogen ion  in the solution
The equivalence of 
 and 
 is one of the logarithm properties we will examine in this section.

### Using the Product Rule for Logarithms

Recall that the logarithmic and exponential functions “undo” each other. This means that logarithms have similar properties to exponents. Some important properties of logarithms are given here.  First, the following properties are easy to prove.
For example, 
 since 
 And 
 since 
 
Next, we have the inverse property.
For example, to evaluate 
 we can rewrite the logarithm as 
 and then apply the inverse property 
 to get 
 
To evaluate 
 we can rewrite the logarithm as 
 and then apply the inverse property 
 to get 
 
Finally, we have the one-to-one property.
We can use the one-to-one property to solve the equation 
 for 
 Since the bases are the same, we can apply the one-to-one property by setting the arguments equal and solving for 
 
But what about the equation 
 The one-to-one property does not help us in this instance. Before we can solve an equation like this, we need a method for combining terms on the left side of the equation.
Recall that we use the product rule of exponents to combine the product of powers by adding exponents: 
 We have a similar property for logarithms, called the product rule for logarithms, which says that the logarithm of a product is equal to a sum of logarithms. Because logs are exponents, and we multiply like bases, we can add the exponents. We will use the inverse property to derive the product rule below.
Given any real number 
 and positive real numbers 
 and 
 where 
 we will show
Let 
 and 
 In exponential form, these equations are 
 and 
 It follows that
Note that repeated applications of the product rule for logarithms allow us to simplify the logarithm of the product of any number of factors. For example, consider 
 Using the product rule for logarithms, we can rewrite this logarithm of a product as the sum of logarithms of its factors:

### Using the Quotient Rule for Logarithms

For quotients, we have a similar rule for logarithms. Recall that we use the quotient rule of exponents to combine the quotient of exponents by subtracting: 
 The quotient rule for logarithms says that the logarithm of a quotient is equal to a difference of logarithms. Just as with the product rule, we can use the inverse property to derive the quotient rule.
Given any real number 
 and positive real numbers 
 
 and 
 where 
 we will show
Let 
 and 
 In exponential form, these equations are 
 and 
 It follows that
For example, to expand 
 we must first express the quotient in lowest terms. Factoring and canceling we get,
Next we apply the quotient rule by subtracting the logarithm of the denominator from the logarithm of the numerator. Then we apply the product rule.

### Using the Power Rule for Logarithms

We’ve explored the product rule and the quotient rule, but how can we take the logarithm of a power, such as 
 One method is as follows:
Notice that we used the product rule for logarithms to find a solution for the example above. By doing so, we have derived the power rule for logarithms, which says that the log of a power is equal to the exponent times the log of the base. Keep in mind that, although the input to a logarithm may not be written as a power, we may be able to change it to a power. For example,

### Expanding Logarithmic Expressions

Taken together, the product rule, quotient rule, and power rule are often called “laws of logs.”  Sometimes we apply more than one rule in order to simplify an expression. For example:
We can use the power rule to expand logarithmic expressions involving negative and fractional exponents. Here is an alternate proof of the quotient rule for logarithms using the fact that a reciprocal is a negative power:
We can also apply the product rule to express a sum or difference of logarithms as the logarithm of a product.
With practice, we can look at a logarithmic expression and expand it mentally, writing the final answer. Remember, however, that we can only do this with products, quotients, powers, and roots—never with addition or subtraction inside the argument of the logarithm.

### Condensing Logarithmic Expressions

We can use the rules of logarithms we just learned to condense sums, differences, and products with the same base as a single logarithm. It is important to remember that the logarithms must have the same base to be combined. We will learn later how to change the base of any logarithm before condensing.

### Using the Change-of-Base Formula for Logarithms

Most calculators can evaluate only common and natural logs. In order to evaluate logarithms with a base other than 10 or 
 we use the change-of-base formula to rewrite the logarithm as the quotient of logarithms of any other base; when using a calculator, we would change them to common or natural logs.
To derive the change-of-base formula, we use the one-to-one property and power rule for logarithms.
Given any positive real numbers 
 and 
 where 
 and 
 we show
Let 
By exponentiating both sides with base
, we arrive at an exponential form, namely 
 It follows that
For example, to evaluate 
 using a calculator, we must first rewrite the expression as a quotient of common or natural logs. We will use the common log.

### Key Equations


### Key Concepts

1. We can use the product rule of logarithms to rewrite the log of a product as a sum of logarithms. See .
2. We can use the quotient rule of logarithms to rewrite the log of a quotient as a difference of logarithms. See .
3. We can use the power rule for logarithms to rewrite the log of a power as the product of the exponent and the log of its base. See , , and .
4. We can use the product rule, the quotient rule, and the power rule together to combine or expand a logarithm with a complex input. See , , and .
5. The rules of logarithms can also be used to condense sums, differences, and products with the same base as a single logarithm. See , , , and .
6. We can convert a logarithm with any base to a quotient of logarithms with any other base using the change-of-base formula. See .
7. The change-of-base formula is often used to rewrite a logarithm with a base other than 10 and 
 as the quotient of natural or common logs. That way a calculator can be used to evaluate. See .

### Section Exercises


### Verbal


### Algebraic

For the following exercises, expand each logarithm as much as possible. Rewrite each expression as a sum, difference, or product of logs.
For the following exercises, condense to a single logarithm if possible.
For the following exercises, use the properties of logarithms to expand each logarithm as much as possible. Rewrite each expression as a sum, difference, or product of logs.
For the following exercises, condense each expression to a single logarithm using the properties of logarithms.
For the following exercises, rewrite each expression as an equivalent ratio of logs using the indicated base.
For the following exercises, suppose 
 and 
 Use the change-of-base formula along with properties of logarithms to rewrite each expression in terms of 
 and 
 Show the steps for solving.

### Numeric

For the following exercises, use properties of logarithms to evaluate without using a calculator.
For the following exercises, use the change-of-base formula to evaluate each expression as a quotient of natural logs. Use a calculator to approximate each to five decimal places.

### Extensions
