Thinking Ahead about Logarithms 1 – You should understand these questions fully before the
next class. Check your answers with the key on your instructor’s website. You can get help
with this work from the following sources:
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Visit your instructor during office hours
Go to the Algebra Alcove in Jones Hall
go to the MAC (the Math Assistance Center) in BH 700
go to http://www.mathsisfun.com/algebra/logarithms.html
It may seem like a strange word, but logarithm is made up of two more familiar words: logos
(Greek for reason or ratio or proportion, as in “logical”) and arithmos (number, as in
“arithmetic”). So logarithm means essentially “proportion number.” A logarithm is an
exponent.
definition of a logarithm:
logbx = n means bn = x.
That base with that exponent produces x.
1. Circle the exponent in the following expressions and equations
a.
b.
c.
d.
e.
f.
35
(-2)3 = -8
42 = 16
(y∙y∙y∙y) = y4
104 = 10,000
bn = x
53 = 125
a logarithm is the power (3) to which a base (5) must be raised to produce a given number (125)
a logarithm is an exponent
2. Write each term in the appropriate blank(s): base, exponent, 3
In the expression, 53, the 5 is called the ____________ and the 3 is called the
________________. In the equation 53 = 125, the logarithm of 125 (with base 5) is _____ .
A logarithm is an _______________ .
When we are given the base, 5, and the exponent, 3, we can evaluate the expression 53:
53 = 5∙5∙5 = 125.
3. Evaluate each expression without using a calculator:
a. 22
b. 23
c. (-2)5
d. 33
e. 62
But what if we are given the base (5) and the answer (125).
Can we go backwards and find out the exponent?
5? = 125
In other words, what is the exponent that will produce the answer?
If we use x instead of a question mark, then we would write
5x = 125
4. Evaluate for x in each expression without using a calculator:
a.
b.
c.
d.
e.
5x = 125 x =
5x = 25 x =
3x = 27 x =
3x = 3 x =
(-2)x = -8 x=
The unknown exponent, x, is “the logarithm,” or the “proportion number.”
We can write this in 2 different ways:
5x = 125 or x = log5 125
When we see or write 5x = 125,
we are writing in the “exponential form” (the logarithm appears as an exponent)
When we see or write x = log5 125,
we are writing in the “logarithmic form” (we are solving for the logarithm x)
We would say “x is the logarithm of 125 with base 5.”
5. Rewrite the following questions as logarithmic equations in the exponential form (5x = 125):
a. To what power do we have to raise 2 in order to produce 8?
b. To what power do we have to raise -3 in order to produce -27?
c. To what power do we have to raise 10 in order to produce 100?
d. To what power do we have to raise 12 in order to produce 144?
6. Now write the same questions in the logarithmic form (x = log5 125):
a. To what power do we have to raise 2 in order to produce 8?
b. To what power do we have to raise -3 in order to produce -27?
c. To what power do we have to raise 10 in order to produce 100?
d. To what power do we have to raise 12 in order to produce 144?