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LOGARITHMS
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.
104 = 10,000
Since
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?
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 inverse of the function logbx is the exponential function bx.
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 0.25 = ?
Answer. 0.25 = ¼ = 2−2. Therefore,
log2 0.25 = log22−2 = −2.
Example 8. log3
Answer.
log3
=?
= 31/5. (Definition of a rational exponent.) Therefore,
= log331/5 = 1/5.
Problem 2. Write each of the following in logarithmic form.
a)
bn = x
logbx = n
b)
23 = 8
log28 = 3
c)
102 = 100 log10100 = 2
d)
5−2 = 1/25.
Problem 3. Write each of the following in exponential form.
a) logbx = n
b) log232 = 5
25 = 32
c) 2 = log864
6−2 = 1/36
d) log61/36 = −2
Problem 4. Evaluate the following.
a) log216
b) log416
c) log5125
d) log81
e) log88
f) log101
d) 1 = log10n
e) logn 1/16 = −2
Problem 5. What number is n?
a) log10n = 3 b) 5 = log2n
f) logn 1/5= −1
c) log2n = 0
g) log2 1/32= n
h)log2 1/2= n
Problem 6. logbbx =
Problem 7. Evaluate the following.
a) log9 1/9=
b) log9 1/81=
c) log2 ¼=
e) log2 1/16=
f) log10 0.01=
g) log10 0.001=
d) log2 1/8=
h) log6
=
i) logb
=
Common logarithms
The system of common logarithms has 10 as its base. When the base is not indicated,
log 100 = 2
then the system of common logarithms - base 10 - is implied.
Here are the powers of 10 and their logarithms:
Powers of 10:
1
1000
1
100
1
10
1
10
100
1000
Logarithms:
−3
−2
−1
0
1
2
3
Logarithms replace a geometric series with an arithmetic series.
Problem 7. log 10n = ?.
Problem 8. log 58 = 1.7634. Therefore, 101.7634 =?
10,000
4
Problem 9. log (log x) = 1. What number is x?
log a = 1, implies a = 10. (See above.) Therefore, log (log x) = 1 implies log x =
10. Since 10 is the base,
x = 1010 = 10,000,000,000
Natural logarithms
The system of natural logarithms has the number called e as its base. (e is named after
the 18th century Swiss mathematician, Leonhard Euler.) e is the base used in calculus.
It is called the "natural" base because of certain technical considerations.
ex has the simplest derivative. Lesson 14 ofAn Approach to Calculus.)
e can be calculated from the following series expressed with factorials:
e=1+
1
1
1
1
+
+
+
+ . . .
1!
2!
3!
4!
e is an irrational number, whose decimal value is approximately 2.71828182845904.
To indicate the natural logarithm of a number we write "ln."
ln x means logex.
Problem 10. What number is ln e ?
Problem 11. Write in exponential form (Example 1): y = ln x.
ey = x.
e is the base.
The three laws of logarithms
1.
logbxy = logbx + logby
"The logarithm of a product is equal to the sum
of the logarithms of each factor."
2.
logb x/y= logbx − logby
"The logarithm of a quotient is equal to the logarithm of the numerator
minus the logarithm of the denominator."
3.
logb xn = n logbx
"The logarithm of a power of x is equal to the exponent of that power
times the logarithm of x."
We will prove these laws below.
Example 9.
Apply the laws of logarithms to log abc²/d 3
Answer. According to the first two laws,
log abc²/d 3=
=log (abc²) − log d 3=log a + log b + log c² − log d 3=log a + log b + 2 log c − 3 log d
Example 12. Solve this equation for x: log 32x + 5 =1
Solution. According to the 3rd Law, we may write
(2x + 5).log 3=1
Now, log 3 is simply a number. Therefore, on distributing log 3,
2x· log 3 + 5 log 3=1
2x· log 3 = 1 − 5 log 3
X= 1 − 5 log 3
2 log 3
By this technique, we can solve equations in which the unknown appears in the exponent.
Problem 12. Use the laws of logarithms to rewrite the following.
a) log ab/c = log a + log b − log c
b) log ab²/c4 = log a + 2 log b − 4 log c
c) log
/z=
d) ln (sin²x ln x)
e) ln
f) ln (a2x − 1 b5x + 1 ) = ln a2x
Problem 13. Solve for x.
(3x + 1) ln 2=5
− 1
+ ln b5x
= (2x − 1) ln a + (5x + 1) ln b
ln 23x + 1= 5
3x ln 2 + ln 2 =5
x=(5 − ln 2)/(3 ln 2)
+ 1
3x ln 2 = 5 − ln 2
−ln x= ln 1/x
Problem 14. Prove:
−ln x = (−1).ln x=ln x−1 Third law = ln 1/x
x=ln 1/x
Proof of the laws of logarithms
The laws of logarithms will be valid for any base. We will prove them for base e, that is,
for y = ln x.
1.
ln ab = ln a + ln b.
The function y = ln x is defined for all positive real numbers x. Therefore there are
real numbers p and q such that
p = ln a and q = ln b.
This implies a = e
p
and b = e q.
Therefore, according to the rules of exponents, ab = e p· e q = ep + q.
And therefore ln ab = ln ep + q = p + q = ln a + ln b.
That is what we wanted to prove.
In a similar manner we can prove the 2nd law. Here is the 3rd:
3.
ln an = n ln a.
There is a real number p such that p = ln a ; that is, a = e p.
And the rules of exponents are valid for all rational numbers n(Lesson 29 of Algebra; an
irrational number is the limit of a sequence of rational numbers). Therefore,
an = e pn. This implies ln an = ln e pn = pn = np = n ln a.
That is what we wanted to prove.
Change of base
Say that we know the values of logarithms of base 10, but not, for example, in base 2.
Then we can convert a logarithm in base 10 to one in base 2 -- or any other base -- by
realizing that the values will beproportional.
Each value in base 2 will differ from the value in base 10 by the same constant k.
Now, to find that constant, we know that
Therefore, on putting x = 2 above:
That implies
Therefore,
That is,
By knowing the values in base 10, we can in this way calculate the values in base 2.
In general, if we know the values in base a, then we can change to base b as follows:
Problem 15. Write the rule for changing from base e to base 8.