Mathematical Investigations III
Name:
Mathematical Investigations III
Logarithms
The First Law of Logarithms
On the same coordinate axes, use a window with x    1,9  and y    3,3  to graph the
following functions. Note that no base is given. In general, when no base is given, assume that
the base is 10.
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f1 x  log 3x
f 2 x  3log x
f3 x  log 3 glog x
f 4 x  log 3  log x
Describe the graphs. Which, if any, are the same?
Repeat the graphs above using the 'LN' function instead of the 'LOG' function.
What changes?
Which graphs are the same?
Is this the same pattern as above?
The 'LN' is the abbreviation for "natural logarithm." This notation indicates that the base of the
logarithm is e ≈ 2.71828 instead of base 10; i.e., LN x  ln x  log e x
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Logs 4.1
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Rev. S03
Mathematical Investigations III
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In fact, it is true that log 3x  log 3  log x . This is true for base 10, base e, or any
other positive base, and the constant 3 is arbitrary.
Let's call this the First Law of Logarithms:
THE FIRST LAW OF LOGARITHMS:
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log b xy = log b x + log b y , b > 0, b ­ 1
We need to prove this law.
Since we know very little about logarithms, let's give names to each part of the right side
of the equation:
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Let log b x  p and log b y  q .
Now, rewrite each of these as something more familiar - exponential equations.
Multiply corresponding sides of these two equations together to form a new equation and
simplify the new equation, using known properties of exponents.
Rewrite your last equation in logarithmic form.
Substitute for p and q.
Compare your results with the box above.
Congratulations! You have just proven the First Law of Logarithms.
Logs 4.2
Rev. S03
Mathematical Investigations III
Name:
Practice:
1.
Rewrite these using the First Law of Logarithms and simplify if possible.
2.
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a.
log b 5b 
b.
log 3 9x 
c.
log5 250  log5 125g2 
d.
log 2 24 
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Prove: log 2 8x  3  log 2 x .
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(Use the method of proof used on page 4.2. Let log 2 x  p and let log 2 8  q .)
Logs 4.3
Rev. S03