Lesson 4: Exponential and Logarithmic Functions
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There are many problems which deal with exponential growth or decay. Exponential functions are those in which the rate of change (either growth positive or decay negative) increase or decrease with time. Exponential functions have the following form: Where f(x) is a function, b is the base and x is the exponent or power. If b is positive than the function continuously increases over time, if x is negative than the function continuously decreases over time. A property of exponential functions is that the slope of the function also continuously increases as x increases. When writing exponential functions it is common to use the carat sign “^”, this sign means raised to the power. Many computer programs and calculators use this sign.	LESSON 4 Printable Lesson
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2 Example of an Exponential Function

Lesson 4: Exponential and Logarithmic Functions
of an Exponential Function
Example In this example lets consider a case where f(x) = 3x Notice we have an exponential function with a base of 3. The values for this function for -3 through 3 would be the following: x -3 -2 -1 0 1 2 3 F(x) 0.03 0.11 0.33 1 3 9 27 The graph of these values would look like the following: For this function as x increases so does y, also as x increases so does the slope of the line. Note that the graph passes through point (0,1), all functions with this form will pass through this point. Another important aspect to point out is that the curve never passes through the x-axis. As we get smaller numbers the graph gets closer and closer to the x-axis but it will never pass through this axis.	LESSON 4 Printable Lesson

3 Logarithmic Functions

Lesson 4: Exponential and Logarithmic Functions
A logarithm is another way to write an exponent. Example If we had the exponent 43 = 64 we could write it as a logarithm as follows: 43 = 64 is the same as log4 64 = 3 Verbally we would say the logarithm of 64 to the base 4 is three. The basic format for a logarithmic function is as follows: Where b is the base of the logarithm. There are two common bases used in mathematics the first is base 10 and the second is e (called the natural log).	LESSON 4 Printable Lesson

4 Logarithmic Operations

Lesson 4: Exponential and Logarithmic Functions
As we just learned logarithms are simply exponents, so as you might expect the same laws or rules apply to performing operations on exponents and logarithms. Here are the basic rules for expanding logarithmic functions: Using these rules you can expand or rewrite logarithmic functions in a way to help you solve them. Example Let’s look at an example: Determine the value of log3 First lets set log3 = to x We now have log3 = x Then we can rewrite the equation such that 3x = Remember that the log function is another way to write an exponent. Now we can reduce to which is equal to 33/4 Therefore log3 = 33/4	LESSON 4 Printable Lesson

5 Logarithms with base 10 and Natural Logarithms

Lesson 4: Exponential and Logarithmic Functions
and Natural Logarithms
As mentioned earlier there are two common bases used in logarithmic functions. The first one is base 10 or common logarthims. This is what you find on your calculator. When we write logarithms to the base 10 we normally do not include the 10. Log10 x means log x The other common base is e, or the natural logarithm. The number e is a irrational constant equal to 2.718 281 … When we write a natural logarithm we use the following: this is the same as writing logex Resource Visit Interactive Mathematics to learn more about where natural logarithms and the number e came from.	LESSON 4 Printable Lesson

6 Solving Exponential and Logarithmic equations

Lesson 4: Exponential and Logarithmic Functions
arithmic Equations
Example Let’s now solve a few problems. Solve 4x = 16 Let’s start by taking the log of each side: log 4x = log 16 Now let’s use one of the log rules I presented before to rewrite the problem as follows: so we get   x log 4 = log 16 Now lets divide both sides by log 4 x = (log 16) / (log 4) This equals = 2 Example Now let’s try one more: Solve for the x in the following: 3ln3 + ln(x - 1) = ln 36 Here are the steps: First let’s simplify 3ln3 using our rules such that you get: ln27 + ln(x -1) = ln 36 This can further be simplified to the following: Ln27(x-1) = ln 36 Now we can divide by the natural log on each side to get the following, which is a simple algebra equation. 27(x-1) = 36 Now divide each side by 27 than add one to each side and you will see that: X = 2.333	LESSON 4 Printable Lesson

7 Review Questions