INDICES AND LOGARITHMS (INDEKS DAN LOGARITMA)
III. Addition/Subtraction and Multiplication/Division
The tables are divided this way:
The rules in the grey area are not "true" index/log laws, but can be used here nonetheless.
Note: Base (English) = Asas (Malay)
ADDITION/SUBTRACTION | |
2 logs of the same base can be combined.
|
Different numbers can only be combined by factorizing. a2 - b2 = (a + b)(a - b) Aside from factorizing, the only other possible way to combine index-forms is when they are EXACTLY the same am +
am
= 2am |
2 logs of different bases must be left alone. log
a
x + log b y |
Numbers with
the same base but different
index must be left alone. (except
when Factorizing)
a5
+ a3 - a2 (The best example is our Good Friend the
Quadratic Equation) Non-logs with the same index but different bases must be left alone. (except when Factorizing) a2
+ b2 + c2 |
MULTIPLICATION/DIVISION |
|
When dividing logs of the same base, they can be combined. (This involves Changing Bases, which is not in the West Australian TEE syllabus)
|
Same
base, different index:
am x
an = am+n Same index,
different base: xa *
ya
= (x * y)a |
NO Logarithms can mix when being multiplied.
Leave them alone.
Logarithms of different bases also do not mix when multiplied/divided. Leave them alone. |
Different
base, different index:
|
Extra
Ok, so we've established the rules for adding/subtracting TWO logs. Now, what happens when you come across something like this?:
- log x a + log x b - log x c + log x d (ax)(by)
Since all of them are the same base, all of them can be combined into 1 logarithm. The general rule is, if the log is positive, it goes up into the numerator; if the log is negative, it goes down into the denominator.
Revisiting the example on the previous page:
log a xy2
It cannot be given a SINGLE coefficient, but can be expanded into 2 logarithms with different coefficients. Using the reverse of the addition/subtraction rule above:
log a (x)(y2) | = log a x + log a y2 |
= log a x + 2log a y |
Summary
Requirements for combining logs:
Requirements for combining numbers in index form:
Summary of summary
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