Name
November 28/29, 2012
Honors Advanced Mathematics
Extra problems (3.5) page 1
Summary of log properties and their proofs:
Property
A. logb x = y  b = x
Possible justifications (you may find others)
Definition of logb(x) function as inverse of bx
(an inverse has the input and output reversed).
B. logb x answers the question
“b to what power equals x?”
This is just a restatement of property A.
C. logb(bx) = x
Apply A to bx = bx or use B (“b to what power equals bx ?”)
D. blogb ( x )  x
Apply A to logb(x)=logb(x)) or use B.
E. logb(rs) = logb(r) + logb(r)
Uses the corresponding exponent property: multiplying
powers with the same base by adding the exponents.
Proof very similar to the proof of E; uses the corresponding
exponent property: dividing powers with the same base by
subtracting the exponents.
Uses the corresponding exponent property: power-to-a-power
by multiplying the exponents.
y
F. logb( rs ) = logb(r) – logb(s)
G. logb(rn) = n logb(r)
H. logb(x) =
log a ( x )
log a (b)
Start with blogb ( x )  x , apply loga( ) to both sides, then
simplify using property G, and finally divide by loga(b).
0. Start by making sure you can prove the basic properties:
a. logb MN  logb M  log b N




b. log b
M
N
 log b M  log b N
c. log b M c  c log b M
d. log b a 
log c a
log c b
Name
November 28/29, 2012
Honors Advanced Mathematics
Extra problems (3.5) page 2
Less well-known properties of logarithms:
Following from the basic log properties, there are various other less-well-known properties.
1. Prove the following facts about logs. You may use the basic properties of logarithms from the table
above.
1
a. loga (b) 
log b (a)
b. loga (b) log b (c) loga (c)

c. loga (b) log b (c) logc (d ) loga (d )

d. loga (x)  k  log b (x) where k is a constant as a function of x

e. log bn ( x n )  log b ( x)



1 
f. log 1   log b ( x)
x 
b
g. a ln b  b ln a