Algebra & Trig, Sullivan & Sullivan, Fourth Edition
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Properties of Logarithms (§4.5):
Product Rule For Logarithms:
Example Of Expanding:
log a xy  log a x  log a y
log 7 5 x  log 7 5  log 7 x
Quotient Rule For Logarithms:
Example Of Expanding:
log a
x
 log a x  log a y
y
log 3
Power Rule For Logarithms:
5
 log 3 5  log 3 7
7
Example Of Expanding:
log a x N  N  log a x 
log 3 x 4  4  log 3 x 
Objective: You want to get practice using these, so later on you can solve logarithmic equations
"Expand" means to keep applying rules, until you can't any more
Example of using more than 1 rule at a time:
log 8 x 3  x  2 
=
<Product Rule>
<Note that we can't use the power rule yet, since the 3 applies ONLY TO X, not the whole
thing>
log 8 x 3  log 8  x  2 
=
<Now we do the power rule>
3log 8 x   log 8 x  2
<At this point, we can't apply any more rules, so we're done expanding>
ICE: Do a couple expansions
Common Errors:
log a x  y   log a x  log a y
log a x  y   log a x  log a y
log a xy  log a x log a y 
 x  log a x
log a   
 y  log a y
We can also go the other way, and un-expand stuff
3 log 10 x  log 10  x  2
ICE: Do some condens-ations

log 10 x 3  log 10  x  2   log 10 x 3 x  2
Algebra & Trig, Sullivan & Sullivan, Fourth Edition
Page 2 / 2
Change of base
So your calculator has a key for log base 10, and probably log base e, but what if you're asked
to find the logarithm of something else? Such as log base 2? Or log base 17?
Use the change of base formula:
log a x 
log b x
log b a , where a & x are given to you, but you get to choose b!
Example:
Find log 17 32 , rounded to 4 significant figures:
log 17 32 
log 10 32 1.505149978
log 10 17  1.230448921  1.2232527111.223
(notice that we keep all the digits until the final, rounding, step)
ICE: Do problems to practice using these