WHEN WE ARE GIVEN the base 2, for example, and exponent 3, then we can evaluate 23.
23 = 8.
Inversely, if we are given the base 2 and its power 8 --
2? = 8
-- then what is the exponent that will produce 8?
That exponent is called a logarithm. We call the exponent 3 the logarithm of 8 with base 2. We write
3 = log28.
We write the base 2 as a subscript.
3 is the exponent to which 2 must be raised to produce 8.
A logarithm is an exponent.
Since
104 = 10,000
then
log1010,000 = 4.
"The logarithm of 10,000 with base 10 is 4."
4 is the exponent to which 10 must be raised to produce 10,000.
"104 = 10,000" is called the exponential form.
"log1010,000 = 4" is called the logarithmic form.
Here is the definition:
logbx = n means bn = x.
That base with that exponent produces x.
Example 1. Write in exponential form: log232 = 5.
Answer. 25 = 32.
Example 2. Write in logarithmic form: 4−2 = 1
16 .
Answer. log4 1
16 = −2.
Problem 1. Which numbers have negative logarithms?
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Do the problem yourself first!
Proper fractions.
Lesson 20 of Arithmetic
Example 3. Evaluate log81.
Answer. 8 to what exponent produces 1? 80 = 1.
log81 = 0.
We can observe that, in any base, the logarithm of 1 is 0.
logb1 = 0
Example 4. Evaluate log55.
Answer. 5 with what exponent will produce 5? 51 = 5. Therefore,
log55 = 1.
In any base, the logarithm of the base itself is 1.
logbb = 1
Example 5 . log22m = ?
Answer. 2 raised to what exponent will produce 2m ? m, obviously.
log22m = m.
The following is an important formal rule, valid for any base b:
logbbx = x
This rule embodies the very meaning of a logarithm. x -- on the right -- is the exponent to which the base b must be raised to produce bx.
The rule also shows that the exponential function bx is the inverse of the function logbx. We will see this in the following Topic.
Example 6 . Evaluate log3 1
9 .
Answer. 1
9 is equal to 3 with what exponent? 1
9 = 3−2
log3 1
9 = log33−2 = −2.
Compare the previous rule.
Example 7. log2 .25 = ?
Answer. .25 = ¼ = 2minus2. Therefore,
log2 .25 = log22−2 = −2.
Example 8. log3Logarithms = ?
Answer. Logarithms = 31/5. (Definition of a rational exponent.) Therefore,
log3Logarithms = log331/5 = 1/5.
Problem 2. Write each of the following in logarithmic form.
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a) bn = x logbx = n b) 23 = 8 log28 = 3
c) 102 = 100 log10100 = 2 d) 5−2 = 1/25. log51/25 = −2.
Problem 3. Write each of the following in exponential form.
a) logbx = n bn = x b) log232 = 5 25 = 32
c) 2 = log864 82 = 64 d) log61/36 = −2 6−2 = 1/36
Problem 4. Evaluate the following.
a) log216 = 4 b) log416 = 2
c) log5125 = 3 d) log81 = 0
e) log88 = 1 f) log101 = 0
Problem 5. What number is n?
a) log10n = 3 1000 b) 5 = log2n 32
c) log2n = 0 1 d) 1 = log10n 10
e) logn 1
16 = −2 4 f) logn 1
5 = −1 5
g) log2 1
32 = n −5 h) log2 1
2 = n −1
Problem 6. logbbx = x
Problem 7. Evaluate the following.
a) log9 1
9 = log99−1 = −1
b) log9 1
81 = −2 c) log2 1
4 = −2
d) log2 1
8 = −3 e) log2 1
16 = −4
f) log10 .01 = −2 g) log10 .001 = −3
h) log6Logarithms = 1/3 i) logbLogarithms = 3/4