Expert II Exponential Functions 30-2 and 30-1

Question 1

How do you use change of base to evaluate an exponent?

Answer 1

Make the large number turn into an exponent with the same base as the other one present in the equation. Then if the bases are the same, the exponents must also be equal. Then evaluate any algebra required.

Question 2

How do you use logarithms to evaluate an exponent?

Answer 2

b^x = a

log_ba = x

Question 3

What is the exponential equation and describe the components.

Answer 3

y = a(b)^x

a is the principal or initial quantity

b is the thing that needs to be multiplied once per calculation (comes from the interest rate but is not necessarily the annual interest rate that is provided, for example if you have 5% per annum in the question, this number might be 1.025 if compounding twice per year)

x is the total amount of calculations needed for the life of the thing (investment, or when you stop calculating)

Question 4

How do you use your graphing calculator to evaluate an exponent by point of intersection?

Answer 4

Set the left side of your equation equal to Y1 and the right side of your equation equal to Y2 and then calculate the point of intersection using CALC.

Question 5

You will need to recall all exponent laws and how to deal with fractional exponents and negative exponents.

Answer 5

For negative recipients be able to come up with this:

3² = 3(3)
3¹ = 3
3⁰ = 1
3^-1 = 1/3
3² = 1/(3(3))

For fractional exponents you need to know that the index becomes the denominator but the exponent is still similarly placed as a numerator.

Question 6

How do you change logarithmic notation into exponential notation?

Answer 6

log_ba = x

b^x = a

Question 7

What are the characteristics of an exponential graph?

Answer 7

Use y = 2^x as the model equation and then apply transformations for any differences.

Also know y = (1/2)^x for any b that is between 0 and 1 exclusive.

Be sure to use a table of values to make both these graphs to determine the:
- asymptote location
- y-int
- general shape

Don’t just do this on your calculator. Make sure you understand where the numbers are coming from and use graphing paper and pencil only.

Then practice writing equations similar to these applying transformations.

Question 8

What are the logarithm rules?

Answer 8

Multiply arguments together when adding two logs with the same base so as to combine them into one log.

Divide arguments when subtracting two logs with the same base so as to combine them into one log.

One log divided by another with the same base can be made into one log.

Remember to use those rules from your formula sheet.

If you do not have your formula sheet with you, then you should be memorizing these.

Change other things into exponential notation when you are not sure how to evaluate them and you will discover rules.

Question 9

What are the characteristics of a logarithmic graph and what exponential equations can get you there?

Answer 9

Do this by hand on paper, switching x and y in your exponential equation, then solving for y by using a logarithm.

Notice that the logarithm ended up creating a situation where x becomes y and y becomes x. So you can reflect over the x = y line.

Then analyze the asymptotes and compare to the exponential equivalents.