Transcript Sec 4.3 Laws of Logarithms

Sec 4.3
Laws of Logarithms
Objective:
•To understand the laws of
logarithms, including the change
of base formula.
Laws of Logarithms
Let a be a positive number, with a ≠ 1.
Let A, B, and C be any real numbers
with A > 0 and B > 0, then the
following laws apply:
loga ( AB )  loga A  loga B
•The logarithm of a product of numbers
is the sum of the logarithms of the
numbers.
Ex. 1 Evaluate
log 4 2  log 4 32
 A
loga    loga A  loga B
B
•The logarithm of a quotient of numbers
is the difference of the logarithms of
the numbers.
Ex. 2 Evaluate
log 2 80  log 2 5
   C log
C
loga A
a
A
The logarithm of a power of a number
is the exponent times the logarithm of
the number.
Ex. 3 Evaluate
1
 log 8
3
Expanding and Combining
Logarithmic Expressions
Ex 4. Use the Laws of Logarithms to
expand each expression.
(a) log2 (6 x )

3
(b) log5 x y
 ab 
(c) ln  3 
 c
6

Ex 5. Combine the following into a single
logarithm.
3 log x + ½ log(x + 1)
Ex 6. Combing the following into a single
logarithm.
3 ln s + ½ ln t – 4 ln(t2 + 1)
Change of Base
loga x
logb x 
loga b
Ex 7. Use the Change of Base Formula and
common or natural logarithms to evaluate
each logarithm, correct to five decimal
places.
(a) log8 5
(b) log9 20
Solving an Exponential Equation
Ex 8. Solve for x. Round to 3 decimal
places.
x 2
3
7
Ex. 9 Solve for x. Round to 3 decimal
places.
8e  20
2x
Ex 10. Solve for x. Round to 3 decimal
places.
3 2 x
e
4
HW – Log Worksheet