Summary Exponentials and Logarithms Notes

Exponentials and Logarithms
1 Logarithms
1.1 Definition
log 𝑘 (𝑎) = 𝑏  𝑎 = 𝑘 𝑏
𝑘 is the base of the logarithm; 𝑏 is the argument; and 𝑎 is the answer (or power).
For example:

log10 (100) = 2  100 = 102
10 is the base; 2 is the argument; 100 is the answer.


1.2 Natural logarithms
Natural logarithms are simply logarithms with base 𝑒. They are most commonly written as ln (𝑥).

ln(𝑥)  log 𝑒 (𝑥)
Applying the definition of a logarithm:

log 𝑒 (𝑥) = 𝑘  𝑥 = 𝑒 𝑘
(Note: whenever you see ln (𝑥), the base is 𝑒; when you see just log (𝑥), with no base indicated, the base
is assumed to be 10.)


1.3 The laws of logarithms
General case Natural logs

log(𝑎𝑏) = 𝑙𝑜𝑔(𝑎) + 𝑙𝑜𝑔(𝑏) ln(𝑎𝑏) = 𝑙𝑛(𝑎) + 𝑙𝑛(𝑏)
𝑎 𝑎
log( ) = 𝑙𝑜𝑔(𝑎) − 𝑙𝑜𝑔(𝑏) ln( ) = 𝑙𝑛(𝑎) − 𝑙𝑛(𝑏)
𝑏 𝑏
log(𝑎𝑛 ) = 𝑛𝑙𝑜𝑔(𝑎) ln(𝑎𝑛 ) = 𝑛𝑙𝑛(𝑎)


I’ve included a separate column for natural logs for clarity, but see it is exactly the same, only the base of
the log in the natural case is 𝑒.
(Note: in the general case there is no base; here, I don’t mean specifically base 10, just logs in general; but,
in any other given problem, if there is no base then do assume the base to be 10.)
1
An extra thought: how can log(𝑎𝑛 ) be written?


1.4 A review of exponential laws
𝑧 𝑎 · 𝑧 𝑏 = 𝑧 (𝑎+𝑏)
𝑧𝑎
= 𝑧 (𝑎−𝑏)
𝑧𝑏