# Exponential and Logarithmic Functions

## Logarithmic Functions


### Learning Objectives

1.  Convert between exponential and logarithmic form. (IA 10.3.1)
2. Evaluate logarithmic functions. (IA 10.3.2)

### Objective 1: Convert between exponential and logarithmic form. (IA 10.3.1)


### Practice Makes Perfect

Since the equations  and  are equivalent, we can go back and forth between them. This will often be the method to solve some exponential and logarithmic equations. To help with converting back and forth, let’s take a close look at the equations. Notice the positions of the exponent and base.
If we remember the logarithm is the exponent, it makes the conversion easier. You may want to repeat, “base to the exponent gives us the number.”

### Practice Makes Perfect


Convert between exponential and logarithmic form.
Remember these logarithmic notations to help complete the following:
 Common Logarithm 

 Natural Logarithm  

### Objective 2: Evaluate logarithmic functions (IA 10.3.2).


We can solve and evaluate logarithmic equations by using the technique of converting the equation to its equivalent exponential form.

### Practice Makes Perfect

In 2010, a major earthquake struck Haiti, destroying or damaging over 285,000 homeshttp://earthquake.usgs.gov/earthquakes/eqinthenews/2010/us2010rja6/#summary. Accessed 3/4/2013.. One year later, another, stronger earthquake devastated Honshu, Japan, destroying or damaging over 332,000 buildings,http://earthquake.usgs.gov/earthquakes/eqinthenews/2011/usc0001xgp/#summary. Accessed 3/4/2013. like those shown in . Even though both caused substantial damage, the earthquake in 2011 was 100 times stronger than the earthquake in Haiti. How do we know? The magnitudes of earthquakes are measured on a scale known as the Richter Scale. The Haitian earthquake registered a 7.0 on the Richter Scalehttp://earthquake.usgs.gov/earthquakes/eqinthenews/2010/us2010rja6/. Accessed 3/4/2013. whereas the Japanese earthquake registered a 9.0.http://earthquake.usgs.gov/earthquakes/eqinthenews/2011/usc0001xgp/#details. Accessed 3/4/2013.
The Richter Scale is a base-ten logarithmic scale. In other words, an earthquake of magnitude 8 is not twice as great as an earthquake of magnitude 4. It is 
 times as great! In this lesson, we will investigate the nature of the Richter Scale and the base-ten function upon which it depends.

### Converting from Logarithmic to Exponential Form

In order to analyze the magnitude of earthquakes or compare the magnitudes of two different earthquakes, we need to be able to convert between logarithmic and exponential form. For example, suppose the amount of energy released from one earthquake were 500 times greater than the amount of energy released from another. We want to calculate the difference in magnitude. The equation that represents this problem is 
 where 
 represents the difference in magnitudes on the Richter Scale. How would we solve for 
 
We have not yet learned a method for solving exponential equations. None of the algebraic tools discussed so far is sufficient to solve 
 We know that 
 and 
 so it is clear that 
 must be some value between 2 and 3, since 
 is increasing. We can examine a graph, as in , to better estimate the solution.
Estimating from a graph, however, is imprecise. To find an algebraic solution, we must introduce a new function. Observe that the graph in  passes the horizontal line test. The exponential function 
 is one-to-one, so its inverse, 
 is also a function. As is the case with all inverse functions, we simply interchange 
 and 
 and solve for 
 to find the inverse function. To represent 
 as a function of 
 we use a logarithmic function of the form 
 The base 
 logarithm of a number is the exponent by which we must raise 
 to get that number.
We read a logarithmic expression as, “The logarithm with base 
 of 
 is equal to 
 ” or, simplified, “log base 
 of 
 is 
 ” We can also say, “ 
 raised to the power of 
 is 
 ” because logs are exponents. For example, the base 2 logarithm of 32 is 5, because 5 is the exponent we must apply to 2 to get 32. Since 
 we can write 
 We read this as “log base 2 of 32 is 5.”
We can express the relationship between logarithmic form and its corresponding exponential form as follows:
Note that the base 
 is always positive.
Because logarithm is a function, it is most correctly written as 
 using parentheses to denote function evaluation, just as we would with 
 However, when the input is a single variable or number, it is common to see the parentheses dropped and the expression written without parentheses, as 
 Note that many calculators require parentheses around the 
 
We can illustrate the notation of logarithms as follows:
Notice that, comparing the logarithm function and the exponential function, the input and the output are switched.  This means 
 and 
 are inverse functions.

### Converting from Exponential to Logarithmic Form

To convert from exponents to logarithms, we follow the same steps in reverse. We identify the base 
 exponent 
 and output 
 Then we write 
 

### Evaluating Logarithms

Knowing the squares, cubes, and roots of numbers allows us to evaluate many logarithms mentally. For example, consider 
 We ask, “To what exponent must 
 be raised in order to get 8?” Because we already know 
 it follows that 
 
Now consider solving 
 and 
 mentally.
1. We ask, “To what exponent must 7 be raised in order to get 49?” We know 
 Therefore, 


2. We ask, “To what exponent must 3 be raised in order to get 27?” We know 
 Therefore, 


Even some seemingly more complicated logarithms can be evaluated without a calculator. For example, let’s evaluate 
 mentally.
1. We ask, “To what exponent must 
 be raised in order to get 
 ” We know 
 and 
 so 
 Therefore, 



### Using Common Logarithms

Sometimes you may see a logarithm written without a base. When you see one written this way, you need to look at the expression before evaluating it. It may be that the base you use doesn't matter. If you find it in computer science, it often means 
. However, in mathematics it almost always means the common logarithm of 10. In other words, the expression 
   often means 


  Currently, we use 
  
     as the common logarithm, 
  
     as the binary logarithm, and 
  
     as the natural logarithm. Writing 
     without specifying a base is now considered bad form, despite being frequently found in older materials.


### Using Natural Logarithms

The most frequently used base for logarithms is 
 Base 
 logarithms are important in calculus and some scientific applications; they are called natural logarithms. The base 
 logarithm, 
 has its own notation, 
 
Most values of 
 can be found only using a calculator. The major exception is that, because the logarithm of 1 is always 0 in any base, 
 For other natural logarithms, we can use the 
 key that can be found on most scientific calculators. We can also find the natural logarithm of any power of 
 using the inverse property of logarithms.

### Key Equations


### Key Concepts

1. The inverse of an exponential function is a logarithmic function, and the inverse of a logarithmic function is an exponential function.
2. Logarithmic equations can be written in an equivalent exponential form, using the definition of a logarithm. See .
3. Exponential equations can be written in their equivalent logarithmic form using the definition of a logarithm See .
4. Logarithmic functions with base 
 can be evaluated mentally using previous knowledge of powers of 
 See  and .
5. Common logarithms can be evaluated mentally using previous knowledge of powers of 
 See .
6. When common logarithms cannot be evaluated mentally, a calculator can be used. See .
7. Real-world exponential problems with base 
 can be rewritten as a common logarithm and then evaluated using a calculator. See .
8. Natural logarithms can be evaluated using a calculator .

### Section Exercises


### Verbal


### Algebraic 

For the following exercises, rewrite each equation in exponential form.
For the following exercises, rewrite each equation in logarithmic form.
For the following exercises, solve for 
 by converting the logarithmic equation to exponential form.
For the following exercises, use the definition of common and natural logarithms to simplify.

### Numeric 

For the following exercises, evaluate the base 
 logarithmic expression without using a calculator.
For the following exercises, evaluate the common logarithmic expression without using a calculator.
For the following exercises, evaluate the natural logarithmic expression without using a calculator.

### Technology

For the following exercises, evaluate each expression using a calculator. Round to the nearest thousandth.

### Extensions 


### Real-World Applications 
