The Logarithm Function
811
In proving the properties of the logarithm function, we first note that many of
the properties involve the quantities log b p and log b q. Let us name these quantities
r and s, respectively, so that
log b p = r and log b q = s
This in turn means that
b
r
= p and b
s
= q
1. Because b > 1, the larger the power to which b is raised, the larger the result.
Thus if p < q, then b
r
< b
s
, so r < s and therefore log b p < log b q.
2. If log b p = log b q, then r = s, so b
r
= b
s
and p = q.
3. log b 1 = 0 because b
0
= 1.
4. log b b = 1 because b
1
= b.
5. log b (b
p
) = p because (translating this equation to its exponential form)
b
p
= b
p
.
6. b
log b p
= p because (translating this equation to its logarithmic form)
log b p = log b p.
7. log b ( pq) = log b p + log b q = r + s because it is true that
b
r+s
= b
r
b
s
= pq, which is the exponential form of the equation we are
trying to prove.
8. log b ( p͞q) = log b p − log b q = r − s because it is true that
b
r−s
= b
r
͞b
s
= p͞q which is the exponential form of the equation we are
trying to prove.
9. log b ( p
q
) = q(log b p) = qr because it is true that b
qr
= (b
r
)
q
= p
q
, which
is the exponential form of the equation we are trying to prove.
10. log a p = log b p͞log b a
Let log a p = w. Then a
w
= p. Now take the logarithm to the base b of both
sides of this equation:
log b (a
w
) = w(log b a) = log b p
or
w = log b p͞log b a
which is the desired result.
11.
p
log b q
= q
log b p
log b ( p
log b q
) = (log b q)(log b p) by Property 9
= (log b p)(log b q)
= log b (q
log b p
) by Property 9
Therefore p
log b q
= q
log b p
by Property 2
The three most useful bases for logarithms are
b = 10 (common logarithm)
b = e, e ∼ 2.7183 (natural logarithm)
b = 2 (what we use throughout this book)