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
log a x + log b x

Numbers with the same base but different index must be left alone. (except when Factorizing)

a5 + a3 - a2
am + an

(The best example is our Good Friend the Quadratic Equation)
x2 + x1


Non-logs with the same index but different bases must be left alone. (except when Factorizing)

a2 + b2 + c2
xm - ym - zm

 

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
am
÷ an = am-n

Same index, different base:
Compare these with the beginning of the previous page

xa * ya = (x * y)a
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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