Page 176 - Calculus Demystified

CHAPTER 6
Transcendental Functions
6.3.2 LOGARITHMS WITH ARBITRARY BASES 163
If you review the first few paragraphs of Section 1, you will find an intuitively
appealing definition of the logarithm to the base 2:
log x is the power to which you need to raise 2 to obtain x.
2
With this intuitive notion we readily see that

log 16 = “the power to which we raise 2 to obtain 16” = 4
2
and

log (1/4) = “the power to which we raise 2 to obtain 1/4” =−2.
2
However this intuitive approach does not work so well if we want to take log 5
√ π
or log 2 7. Therefore we will give a new definition of the logarithm to any base
a> 0 which in simple cases coincides with the intuitive notion of logarithm.
If a> 0 and b> 0 then
ln b
log b = .
a
ln a
EXAMPLE 6.19
Calculate log 32.
2
SOLUTION
We see that

ln 32 ln 2 5 5 · ln 2
log 32 = = = = 5.
2
ln 2 ln 2 ln 2
Notice that, in this example, the new definition of log 32 agrees with the
2
intuitive notion just discussed.
EXAMPLE 6.20
Express ln x as the logarithm to some base.
SOLUTION
If x> 0 then
ln x ln x
log x = = = ln x.
e
ln e 1
Thus we see that the natural logarithm ln x is precisely the same as log x.
e
Math Note: In mathematics, it is common to write ln x rather than log x.
e
You Try It: Calculate log 27 + log (1/25) − log 8.
5
3
2