1. Logarithms: “undoing” exponents

2. Recap Last week we looked at RATIONAL exponents and saw that
A square root is the same as an exponent of ½
A cubed root is the exponent 1/3
To evaluate powers with rational exponents, we “rip the exponent apart”.
We also saw that radioactive materials will decay in an exponential fashion (half-life)
We also saw that compound interest can be modeled using exponential equations

3. Compound Interest Formula
Where
A is the $ amount in the account at time t (years) P is the principle (initial) $ amount (when t = 0) i is the decimal value of the annual interest rate n is how many times per year the interest is compounded t is the number of years
Look for terms like: daily (n = 365), weekly (n = 52) quarterly (n = 4) semi-annually (n = 2) monthly (n = 12)

4. More Compound Interest Examples
Ex 1. A credit card charges 24.2% interest per year compounded monthly. There are $900 worth of purchases made on the card. Calculate the amount owing after 18 months. (Assume that no payments were made.)
P = 900
i = 24.2% = 0.242
n = 12
t = 1.5
y = ?

5. More Compound Interest Examples
Ex 2. A bank account earns interest compounded monthly at an annual rate at 4.2%. Initially the investment was $400. When does it double in value?
So this questions seems to be like all the others…
P = $400
i = 4.2% = 0.042
n = 12
t = ?
y = $800
And now we get a common base…
…except we can’t.
…maybe by the end of class…

6. Logarithmic Form
We`re totally stuck. So far we can only solve for an exponent when we can achieve common bases. Good thing we have logarithms to help.
exponent
exponent
can also be written as
base
argument
base
argument
That is “log base 2 of 8 equals 3”
This form is helpful because the exponent value is no longer locked in the exponent position, in fact it is isolated.

7. Going from exponential to logarithmic form
Write the following exponential equations in logarithmic form.
When the base is 10, we usually leave it out… take a look at your calculator…

8. Going from logarithmic to exponential form
Write the following logarithmic equations in exponential form.

9. Evaluating Logs By changing forms we can evaluate log expressions.
Examples
or, solve for x:
a) Evaluate:
This asks “2 to the what gives 32?”
We know this is 5, so:
b) Evaluate:
or, solve for x:
This asks “4 to the what gives 64?”
We know this is 3, so:

10. Evaluating Logs c) Evaluate: or, solve for x:
This asks “1/4 to the what gives 32?”
We can get common bases :
or, solve for x:
d) Evaluate:
We know this is 2, so:

11. Solving log equations
Solving logarithmic equations takes some instinct, which only comes from practice, but to help you get you started, here is a flowchart with some possibly useful steps.
Simplify: evaluate any complete log or exponential expressions
Isolate the unknown:
If the unknown is in the…
…argument: change it to exponential form
…exponent: in exponential form get common bases if possible, or change to logarithmic form to solve, or take the log of both sides and apply log rules
…base: write in exponential form then remove the exponent by raising each side to the opposite exponent

12. Solving log equations - Examples
b) Solve for x.
a) Solve for x.
Unknown in the exponent
Unknown in the argument

13. Solving log equations
Unknown in the base c)

14. Solving log equations Unknown in the argument d) 10
But wait… the base here is...
Since the calculator uses base 10, just type this in and get…

15. Solving log equations - Practice
Solve each equation for x a. b. c. d. e. f. g. h.

16. Solving log equations - Practice
Evaluations
Solutions

17. Solving log equations - Practice
Solutions
Evaluations

18. Solving log equations - Practice
Solutions
Evaluations

19. Solving log equations - Practice
Solutions

20. Solving log equations - Practice
Solutions

21. Solving log equations - Practice
Solutions
Evaluations

22. Solving log equations - Practice
Solutions
We don’t know what is.
But notice that there’s a common base on both sides of the equation.
Since the bases are equal, the ARGUMENTS must be equal.

23. The first law of logarithms
Remember the laws of exponents:
First law of logarithms:
When multiplying powers with the same base, we keep the base and add the exponents.
When adding logs with the same base, we keep the log and base and multiply the arguments
Let and
So and

24. The second law of logarithms
Remember the laws of exponents:
Second law of logarithms:
When dividing powers with the same base, we keep the base and subtract the exponents.
When subtracting logs with the same base, we keep the log and the base and divide the exponents.
Let and
So and

25. Laws of Logarithms - Practice
Practice with the first two logarithm laws. Solve for x.

26. The third law of logarithms
Remember the laws of exponents:
Third law of logarithms:
The “down in front” rule
When we have a power of a power, we keep the base and multiply the exponents.
If the argument of a logarithm is a power, the exponent can be moved “down in front”.
Let So

27. Laws of Logarithms - Practice
Practice with the third logarithm law.
Evaluate.
Solve.

28. Shortcut for evaluating logs
This third law of logs is the key to evaluating logs with bases other than 10 on your calculator!
Let
change to exponential form
take the log of both sides
apply the “down in front” rule.
remember that
Logarithm Shortcut 1: To evaluate logab on your calculator, divide logb by loga.
Remember, the base is on the bottom

29. Shortcut for solving for exponents
This third law of logs is the key to solving exponential equations when common bases can’t be achieved!
take the log of both sides
apply the “down in front” rule.
Logarithm Shortcut 2: To solve for an exponent, divide the log of the argument by the log of the base.
Remember, the base is on the bottom

30. Laws of Logarithms - Practice
Practice with the third logarithm law.
Evaluate.
Solve.
Solve again.
Evaluate again.

31. Laws of Logarithms – Practice
Solve for x.

32. Another shortcut for evaluating certain logarithms
logb(ba) = a
Examples
Challenge: Can you prove why this shortcut is true? logb(ba) = a

33. And back to the beginning… We can now solve this question!
Ex 2. A bank account earns interest compounded monthly at an annual rate at 4.2%. Initially the investment was $400. When does it double in value?
So this questions seems to be like all the others…
P = $400
i = 4.2% = 0.042
n = 12
t = ? years
y = $800
We can’t get common bases… but…..