Download Section 6.4

MATH 1100
SECTION 6.4 Notes
Laws of Logarithms – Text Pages 407-411
Laws of Logarithms:
Let a be a positive number, with a  1 . Let A  0 , B  0 and C be
any real numbers.
1.)
The logarithm of a product of numbers is the sum of the
logarithms of the numbers:
log a  AB  log a A  log a B
2.)
The logarithm of a quotient of numbers is the difference of the
logarithms of the numbers:
 A
log a    log a A  log a B
B
3.)
The logarithm of a power of a number is the exponent times the
logarithm of the number:
 
log a AC  C  log a A
WARNING:
The Laws of Logarithms show us how to compute logarithms for a
product, quotient, or a power. However, there is NO corresponding
rule for the logarithm of a sum or difference!!
log a x  y   log a x  log a y .
So,
Also, be careful not to incorrectly use the quotient and power
rules:
log 6
6
 log  
log 2
2
and
log 2 x 3  3  log 2 x
Change of Base Formula:
log a x
log a b
***This formula is particularly useful when using your calculator since it
is programmed only to calculate the common logarithm (base 10) and the
natural logarithm (base e)***
log b x 
Example 1:
Using the Laws of Logarithms, rewrite the expression
in a form with no logarithm of a product, quotient or power.
(a.)
log 4 xz 
(b.)
 y
log 4   
x
(c.)
log 4
(d.)
 y 5 w2 
log a  4 3  
x z 
(e.)


z4 x

ln 
 3 y 2  6 y  17 


 z
3
Example 2:
Rewrite the expression as a single logarithm.
(a.)
1
2 log a x  log a x  2  5 log a 2 x  3 
3
(b.)
1
4 log x  log x 2  1  2 log x  1 
3
(c.)
ln 5  2 ln x  3 ln x 2  5




Example 3:
Evaluate the expression,
log 4 192  log 4 3
Example 4:
Use the change of base formula and a calculator to evaluate the
logarithm, correct to six decimal places.
(a.)
log 3 11 
(b.)
log 4 322 
(c.)
log 7 21 