1. 5.5 Properties and Laws of Logarithms
Do Now: Solve for x.
x = 3
x = 1/3
x = 6
x = 12

2. 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

3. 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

4. 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! **

5. Example 1: Solving Equations Using Properties
Use the basic properties of logarithms to solve each equation.

6. Laws of Logarithms aman=am+n
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.

7. Product and Quotient Laws of Logarithms
For all v,w>0,
log(vw) = log v + log w
ln(vw) = ln v + ln w

8. Using Product and Quotient Laws
Given that log 3 = and log 4 = , find log 12.
Given that log 40 = and log 8 = , find log 5.
log 12 = log (3•4) = log 3 + log 4 =
log 5 = log (40 / 8) = log 40 – log 8 =

9. 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

10. Using the Power Law Given that log 25 = 1.3979, find log .
Given that ln 22 = , find ln 22.
log (25¼) = ¼ log 25 =
ln (22½) = ½ ln 22 =

11. 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) =

12. Simplifying Expressions
Simplify each expression.
log 8x + 3 log x – log 2x2
log 4x2