Math 9

Week 2: Exponential & Logarithm Lesson

Fall 2008

Exponential Functions: An exponential equation contains a variable as an exponent. Examples:

1)

2a = 25

2)

9 x = 81

3)

7 x −3 =

4)

4 x = 82 x + 4

5)

6x = 5

1 49

In Example 1 the only way these two terms can be equal is if ____ = _____ . In Example 2 the base numbers are not the same. Sometimes we can solve an exponential equation by rewriting each side of the equation in terms of the same base number. Since 81 = 9 2 , we can rewrite the equation as: 9 x = ________ . Then we can see that x =_____ . In Example 3, we need to use some of the exponent laws that we reviewed last week in order to rewrite the equation in terms of the same base number.

1 can be written as _________ using Exponent Law #6. 49 Now Example 3 can be rewritten as: 7 x −3 = ________ and then we can solve for x. In this case, x = _________ . In Example 4, we have to change both sides of the equation in order to get the same base number. 8 is not a power of 4, but both 8 and 4 are powers of 2! Rewrite 4 x and 82 x + 4 to be powers of 2 using Exponent Law #2. Then solve for x. _____x = ______2x+4

x = _________ In Example 5 none of our current tools will work to solve this equation. We need some new tools!! 1

Before we learn about the tools that will allow us to solve problems like Example 5, let’s look at the graphs of exponential functions. A. Make a table of values for the exponential function: y = 2 x and graph those values.

x

y = 2x

–3 –2 –1 0 1 2 3

What would happen if the exponential function was negative? Let’s see. B. Fill in the table and then graph the points for the function: y = −2 x

x

y = −2x

–3 –2 –1 0 1 2 3 2

What if the negative is in the exponent? What will that do to the graph? C. Fill in the table and graph the points for the function: y = 2− x

x

y = 2− x

–3 –2 –1 0 1 2 3

What do you notice about these graphs of exponential functions? 1. 2. 3.

3

An inverse function is when the x and y values are switched. When we do this to y = 2 x , the y values from our table become the x values and the x values become the new y values. The new table looks like this: Fill in the missing values. D.

x

y –3 –2

1 2

2

1

4 3

How does this compare with the graph of y = 2 x which you did earlier??

This brings us to the definition of logarithms.

Definition of Logarithms:

If b x = a, then logb a = x. If logb a = x, then a = b x . This can also be written as: b x = a ⇔ logb a = x

When we compare the graphs of A (y = 2x) and D (x = 2y or y = log2x), we see that the graphs of inverse functions keep the same shape but are flipped. (They are actually reflected about the line y = x). 4

x From the definition of logarithms and using a calculator, we can solve problems like: 10 = 7

First rewrite this equation in its logarithmic equivalent. (When using base 10 logarithm— log10—it is without the base and is called the “common logarithm”). log7 = x A scientific calculator has two logarithmic functions: base 10 and base e. The button that says “log” is base 10. (the one that says “ln” is base e). So to solve this equation, find log 7 on your calculator and give your answer to 5 decimal places. Thus, log7 ≈ ________________ . The beginning of logarithms and their uses. Logarithms were invented independently in the early 1600s by John Napier in Scotland and by Joost Burgi in Switzerland. They were used to simplify mathematical calculations—remember there were no calculators than other than pencil and paper at that time. They both invented what we now call Common Logarithms—meaning the base, b, was always 10. Logarithms are still used in many fields today. Some of these include: finance, astronomy, interest rates, population growth and decay, radioactive growth and decay, and earthquake magnitude calculation.

Properties of Logarithms: b, M, N > 0 1. logb 1 = 0

since b0 = 1

Example: 30 = _______

2. logb b = 1 since b = b

Example: log8_____ = 1

1

3. logb b = x since b = b x

4.

b(

logb x )

x

x

Example: log773 = _________

= x since logb x = ( logb x )

where x>0

5. Product Law: logb ( M ⋅ N ) = logb M + logb N Example:

log7 ( 49 ⋅ 7 ) = log7 49 + log7 7 = log7 72 + log7 7

= _____ + _____ by Properties 3 & 2 = ______ Example:

log5 3x = _____________________________ 5

6. Quotient Law:

Example:

7. Power Law: Example:

⎛M ⎞ logb ⎜ ⎟ = logb M − logb N ⎝N⎠ ⎛4⎞ log3 ⎜ ⎟ = ⎝ x ⎠ _____________________________

logb M k = k ⋅ logb M log4 x 3 = __________________

8. Equality Law: If two logs (of the same base) of two numbers (called “arguments”) are equal, then the numbers (arguments) are equal.

Example:

log3 2 x = log3 14 ( Here 2x and 14 are the arguments of the log ) 3 2x = 14 x=7

9. Changing bases algorithm (usually in order to use a calculator): a. If given y = logb x , first rewrite in exponential form. __________________ b. Take the common log of both sides: c. Use the Power Law (or other Property):

____________________ _____________________ ⎛x⎞ d. Solve for y: _______________ Note: this is NOT logb ⎜ ⎟ !! ⎝b⎠ e. Use your calculator to find an approximation of the answer to a specified number of decimal places.

Example: a.

Find

log3 4 = x

to 4 decimal places.

3x = 4

b. log3 x = log 4 c. ____________ = log 4 d. x = ________________ e. x ≈ _______________ 6

Here are some practice problems.

log2 x = 3

1.

Solve for x:

2.

Complete the table below: Logarithmic Form

Exponential Form

Log3 81 = 4 Log4 1 = 0 91 = 9 1 = 0.01 102 3.

4.

Using the Properties of Logarithms, expand the following expressions: a)

log3 (9x)

b)

⎛x⎞ log ⎜ ⎟ ⎝5⎠

c)

⎛6⎞ log6 ⎜ ⎟ ⎝x⎠

d)

log3 x5

e)

log3 ( 27x )

2

Using the Properties of Logarithms, simplify the following expressions as much as possible: a)

log3 4 – log3 5

b)

log4 4 + log4 16

c)

log3 4 – (log3 5 + log3 12)

5.

Solve for x: log3 27 = 3x

6.

Solve for x: log2 (4) + log2 (x + 1) = 3

7.

Solve for x: log 10000 = x

7