INTERESTING QUESTIONS

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INTERESTING QUESTIONS UNDOING EXPONENTS LOGS

• Hey, hey logarithmWhen we get the blues

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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
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InterestInterest
In general, the compound interest is

• Where
• A is the amount in the account at time, t
• P is the principle (initial) amount
• r is the decimal value of the interest rate
• n is how many times per year the interest is
  compounded.

Look for terms like daily (n 365),
semi-annually (n 2), weekly (n 52) and
monthly (n 12)
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InterestInterest
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.)
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A new type of question?
Ex 4. 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
And now we get
Common bases
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Solving for the Exponent
Were totally stuck! We presently have no way of
solving for an exponent unless we can get common
bases.
exponent
exponent
So mathematicians invented logarithms.
can also be written as
base
base
argument
argument
We read this log to base 2 of 8 equals 3.
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Going from one form to another
Write the following in logarithmic form.
When the base is a 10 we do not need to write
it. This is because base 10 is what most
calculators deal with.
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Going from one form to another
Write the following in exponential form.
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Evaluating Logs
By changing forms we can evaluate log expressions.
Evaluate a)
This asks 2 to the what gives 32?
or
We know this is 5.
b)
This asks 4 to the what gives 64?
or
We know this is 3.
c)
This asks 1/4 to the what gives 32?
We can get common bases
d) log 100
This asks 10 to the what gives 100?
We know this is 2.
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Solving log equations common bases
To continue getting used to logs, well look at
these questions.
Solve for x.
Were stuck in log form so go to exponential form
a)
We can solve this x 1/8
Were stuck in log form so go to exponential form
b)
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Solving log equations common bases
c)
Were stuck in log form so go to exponential form
d)
Were stuck in log form so go to exponential form
To solve for the base, we can undo the exponent
by raising both sides to the 5/4.
But wait the base here is
10
Since the calculator uses base 10 I just type
this in and get
log1.30.114
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Try some
1) Solve each equation
a.
a. b. c. d. e. f. g. h.
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Try some
1) Solve each equation
b.
a. b. c. d. e. f. g. h.
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Try some
1) Solve each equation
c.
a. b. c. d. e. f. g. h.
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Try some
1) Solve each equation
d.
a. b. c. d. e. f. g. h.
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Try some
1) Solve each equation
e.
a. b. c. d. e. f. g. h.
Since I cant get common bases, Im stuck in
exponential form. So I go to log form.
My calculator can find this.
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Try some
1) Solve each equation
f.
a. b. c. d. e. f. g. h.
Im stuck in log form. So I go to exponential
form.
To solve for the base, I undo the exponent. I
raise both sides to the -3/2
The negative in the exponent means I flip the
base
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Try some
1) Solve each equation
g.
a. b. c. d. e. f. g. h.
Im stuck in log form. So I go to exponential
form.
Rip the exponent apart
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Try some
1) Solve each equation
h.
a. b. c. d. e. f. g. h.
I dont know what is.
But I do notice that theres a common base on
both sides of the equation.
Since the bases are equal, the ARGUMENTS must be
equal.
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So What?
We can now write an equation where we solve for
the exponent but how do we evaluate the exponent
if we do not recognize the log or if we cannot
get common bases?
All of these started with the equation
Are we any closer?
Well we can write it in a log form
But thats as far as we can get.
How do the laws of exponents relate to logs?
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The Laws of Logs
Remember the laws of exponents
When multiplying powers with the same base, we
keep the base and add the exponents.
Let
Go to log form
When adding logs with the same base, we keep the
log and base and multiply the arguments.
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The Laws of Logs
Remember the laws of exponents
When dividing powers with the same base, we keep
the base and subtract the exponents.
Let
Go to log form
When subtracting logs with the same base, we keep
the log and base and divide the arguments.
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Laws of Exponents
Practice with the first two laws. Solve for x.
Im stuck in log form. So I go to exponential
form.
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The 3rd Law of Logs
When we have a power of a power, we keep the base
and multiply the exponents.
Let
Raise both sides to the exponent b
The down in front rule!
Ex. Evaluate
Go to log form
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And finally
Are we any closer to solving the original
question?
Lets take the log of both sides
Now the down in front rule
Divide by 12log1.0035
And my calculator can do this
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A shortcut to the calculator rule
So we have seen that
can be written as
So we do not need to take the log of both sides.
We can go to log form
And then write
Remember that the base is on the bottom!
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Lots o Logs
Solve for x.
Since I cant get common bases, Im stuck in
exponential form. So I go to log form.
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Lots o Logs
Solve for x.
Since I cant get common bases, Im stuck in
exponential form. So I go to log form.
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Lots o Logs
Solve for x.