LESSON SIX: Logarithms
Recall: Inverse operations: adding/subtracting, multiplying/dividing, etc…
There are also INVERSE FUNCTIONS (or relations).
Consider the function y  x 2 . The inverse is found by interchanging x and y.
x  y2
now, solve for y by " square rooting " both sides
 xy
or y   x
Now consider the function y  2 x . If we wanted to find the inverse, we would still interchange x and y,
but then there would be a problem to solve for (or isolate) y.
x  2y
So, a new notation was invented called LOGARITHMS. The “log” function is the inverse of the
exponential function.
x  2 y is equivalent to y  log 2 x
We read “ y is the exponent we put on the base 2 to get x.”
Example: Write what each of the following mean in sentence form.
a) y  log 3 k
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b) h  log 5 t
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c)
4  log 3 81
d) 2  log 10 100
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Note: Since base 10 is used in many applications, when we write, for example, log 1000 , we mean base 10.
i.e., if no base is written, it means base 10.
So, the previous example could be written as 2  log 10 100 or 2  log 100 and it would mean the same
thing.
More examples: 1. Write the following in exponential form.
a) log 4 64  3
b) log 9 729  3
c) log 1000  3
d) log 2 1024  10
2. Write the following in logarithmic form.
a) 35  243
b) 6 4  1296
c) 5 3  125
d) 10 4  10,000
Note: Your calculator can only evaluate logarithms with base 10 or base e (don’t worry about base e for
now.) All other logarithms will have to be evaluated without the aid of a calculator.
Hw:
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Handout “P.227” #1-5
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Graph y  log 2 x on the grid below by reflecting y  2 x in the line y=x.
(Pick a few ordered pairs and interchange the x- and y- values. When you are done, show me and I will
give you a mark for it.)