5.5 Properties and Laws of Logarithms
Do Now: Solve for x.
1. log 4 64   x
2. log 27 3   x
x=3
x = 1/3

  x
3. log 10
6
 x
x=6
4. ln e
12
x = 12
Consider some more examples…
Without evaluating log (678), we know the
expression “means” the exponent to which
10 must be raised in order to produce 678.
log (678) = x  10x = 678
If 10x = 678, what should x be in order to
produce 678?
x = log(678) because 10log(678) = 678
And with natural logarithms…
Without evaluating ln (54), we know the
expression “means” the exponent to which e
must be raised in order to produce 54.
ln (54) = x  ex = 54
If ex = 54, what should x be in order to
produce 54?
x = ln(54) because eln(54) = 54
Basic Properties of Logarithms
Common Logarithms
Natural Logarithms
1. log v is defined only when
v > 0.
1. ln v is defined only when v > 0.
2. log 1 = 0 and log 10 = 1
2. ln 1 = 0 and ln e = 1
3. log 10k = k for every real
number k.
3. ln ek = k for every real number k.
4. 10logv=v for every v > 0.
4. elnv=v for every v > 0.
** NOTE: These properties hold for all bases –
not just 10 and e! **
Example 1: Solving Equations Using Properties
Use the basic properties of logarithms to solve
each equation.
1. log(x  3)  5
10
log(x  3)
 10
x  3  10
5
5
5
x  10  3
x  100,003
2. ln(2x  1)  7
e
ln(2x  1)
e
7
2x  1  e
7
7
2x  e  1
x 
7
e 1
2
Laws of Logarithms
Because logarithms represent exponents, it is helpful to
review laws of exponents before exploring laws of
logarithms.
When multiplying like bases, add the exponents.
aman=am+n
When dividing like bases, subtract the exponents.
a
m
a
n
a
m n
Product and Quotient Laws of Logarithms
For all v,w>0,
log(vw) = log v + log w
ln(vw) = ln v + ln w
For all v, w  0,
log  wv   log v  log w
ln  wv   ln v  ln w
Using Product and Quotient Laws
1. Given that log 3 = 0.4771 and log 4 = 0.6021,
find log 12.
log 12 = log (3•4) = log 3 + log 4 = 1.0792
2. Given that log 40 = 1.6021 and log 8 = 0.9031,
find log 5.
log 5 = log (40 / 8) = log 40 – log 8 = 0.6990
Power Law of Logarithms
For all k and v > 0,
log vk = k log v
ln vk = k ln v
For example…
log 9 = log 32 = 2 log 3
Using the Power Law
1. Given that log 25 = 1.3979, find log
4
25
log (25¼) = ¼ log 25 = 0.3495
2. Given that ln 22 = 3.0910, find ln 22.
ln (22½) = ½ ln 22 = 1.5455
.
Simplifying Expressions
Logarithmic expressions can be simplified using logarithmic
properties and laws.
Example 1:
Write ln(3x) + 4ln(x) – ln(3xy) as a single logarithm.
ln(3x) + 4ln(x) – ln(3xy) = ln(3x) + ln(x4) – ln(3xy)
= ln(3x•x4) – ln(3xy)
= ln(3x5) – ln(3xy)
= ln
 
= ln
 
5
3x
3xy
4
x
y
Simplifying Expressions
Simplify each expression.
1. log 8x + 3 log x – log 2x2
2. ln
   ln 
x
x
4
ex
2

log 4x2
¼