Home >> PURE MATHS, Algebra, laws of logarithms
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The Laws of Logarithms
Proofs #1
prove that 
let 
(i
(ii
then 
it follows that 
taking logs to the base 'a' each side,
but
therefore 
substituting for A and B from (i and (ii
Proofs #2
prove that 
let 
which implies that 
taking logs on both side to the base 'b'
rearranging to make 'y' the subject
substituting for 'y' ( )
Changing the Base
Remember that the change of base occurs in the term where the base is 'x' or some other variable.
Example #1
solve for x 
changing 
to the base '2'
multiplying both sides by 
rearranging 
remembering that 
factorising the quadratic
giving roots
(implying that)
Simultaneous Equations
'Substitution' simultaneous equations are common problems.
method:
| 1. first find what x is in terms of y |
| 2. then substitute for x in the other equation |
| 3. solve for y |
Example #1
given that 
(i
and 
(ii
find x and y
implies that 
but 
(iii
substituting for x into (ii
(implying that)
answer 
answer 
Variable in the Index
method:
1. take logs on both sides |
2. move the indices infront of the logs |
3. expand the equation |
4. collect x-terms to the left |
| 5. sum the numbers to the right |
These problems can be tricky with the amount of arithmetic involved.
So make sure you write everything down to make checking your working easier.
Example #1
solve for x to 3 d.p. 
taking logs to base 10 on each side
expanding the powers
substituting the values of logs to base 10 for 2, 3 and 6
expanding,
collecting terms,
to 3 d.p.
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