5.1
LOGARITHMS
AND THEIR PROPERTIES
1
What is a Logarithm?
If x is a positive number, log x is the
exponent of 10 that gives x.
In other words, if
y = log x
then 10y = x.
log 100 = 2
because
log 2500 ≈ 3.398 because
102 = 100
103.398 = 2500
2
Logarithms to Exponents
Example 1
Rewrite the following statements using exponents instead
of logs.
(a) log 100 = 2
(b) log 0.01 = −2
(c) log 30 = 1.477
Solution
We use the fact that if y = log x then 10y = x.
(a) 2 = log 100 means that 102 = 100.
(b) −2 = log 0.01 means that 10−2 = 0.01.
(c) 1.477 = log 30 means that 101.477 ≈ 30.
3
Exponents to Logarithms
Example 2
Rewrite the following statements using logs instead of
exponents.
(a) 105 = 100,000 (b) 10−4 = 0.0001 (c) 10.8 ≈ 6.3096.
Solution
We use the fact that if 10y = x, then y = log x.
(a) 105 = 100,000 means that log 100,000 = 5.
(b) 10−4 = 0.0001 means that log 0.0001 = −4.
(c) 10.8 = 6.3096 means that log 6.3096 ≈ 0.8.
4
Logarithms Are Exponents
Example 3
Without a calculator, evaluate the following, if possible:
(a) log 1
(c) log 1,000,000
(d) log 0.001
(f) log(−100)
Solution
(a) We have log 1 = 0, since 100 = 1.
(c) Since 1,000,000 = 106, the exponent of 10 that gives
1,000,000 is 6. Thus, log 1,000,000 = 6.
(d) Since 0.001 = 10−3, the exponent of 10 that gives 0.001 is
−3. Thus, log 0.001 = −3.
(f) Since 10 to any power is positive, −100 cannot be written
as a power of 10. Thus, log(−100) is undefined.
5
Logarithmic and Exponential
Functions are Inverses
For any N,
log(10N) = N
and for N > 0,
10logN = N.
log(108.5 )  8.5
10log2.7  2.7
10log( x3)  x  3
6
Properties of Logarithms
Properties of the Common Logarithm
• By definition, y = log x means 10y = x.
• In particular,
log 1 = 0
and
log 10 = 1.
• The functions 10x and log x are inverses, so they “undo” each
other:
log(10x) = x for all x,
10log x = x for x > 0.
• For a and b both positive and any value of t,
log(a b) = log a + log b
log(a/b) = log a − log b
log(bt) = t · log b.
7
Applying Properties of Logarithms
Example 5
Solve 100・2t = 337,000,000 for t.
Solution
Dividing both sides of the equation by 100 gives
2t = 3,370,000.
Taking logs of both sides gives
log 2t= log(3,370,000).
Since log(2t) = t · log 2, we have
t log 2 = log(3,370,000),
so, solving for t, we have
t = log(3,370,000)/log 2 = 21.684.
8
The Natural Logarithm
For x > 0,
ln x is the power of e that gives x
or, in symbols,
ln x = y means ey = x,
and y is called the natural logarithm of x.
9
Same Properties as for Common Logarithm
Properties of the Natural Logarithm
• By definition, y = ln x means ey = x.
• In particular,
ln 1 = 0
and
ln e = 1.
• The functions ex and ln x are inverses, so they “undo” each
other:
ln(ex) = x for all x,
eln x = x for x > 0.
• For a and b both positive and any value of t,
ln (a b) = ln a + ln b
ln (a/b) = ln a − ln b
ln (bt) = t · ln b.
10
Applying Properties of Natural Logarithms
Example 6 (a)
Solve for x: 5e2x = 50
Solution (a)
We first divide both sides by 5 to obtain
e2x = 10.
Taking the natural log of both sides, we have
ln(e2 x )  ln10
2x  ln10
ln10
x
 1.151
2
11
Misconceptions and Calculator Errors
Involving Logs
NO NO NO
• log(a + b) is not the same as log a + log b
• log(a − b) is not the same as log a − log b
• log(a b) is not the same as (log a)(log b)
• log (a/b) is not the same as (log a) / (log b)
• log (1/a) is not the same as 1 / (log a).
12