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Online Quiz (Worksheet A B C D)

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High School Mathematics
3.24 Change of base of a Logarithm

log x a = log y a . log x y.
Proof:
Let log x a = p, Then xp = a -------I
log y a = r, then yq = a -----------II
also let log x y = r, then y = xr --III
From I and II we get, a = xp = yq ---IV
From III, we get, y q = (xr)q = xrq ---V
xp =xrq
bases are equal and so indices are also equal.
Therefore, p = rq
log x a = log y a . log x y.
This result is used to change the base of the logarithm.

Note:
log x a = log y a . log x y.
in the above equation put x = a and y = b, we get.
log a a = log b a . log a b.
log b a . log a b = 1. [Since log a a = 1]

Example:
Show that log y x . log z y . log x z = 1.
Solution:
logy x . log z y = log z x [Since logy a . log x y = log x a]
= logz x . log x z
= logx x [Since logy a . log x y = log x a]
= 1 [Since log a a = 1]


Directions: Prove the following statements. Also write at least ten examples of your own.

1. Show that, log 3 2 . log 4 3 . log 2 4 = 1.

2. Show that, log b a . log c b . log a c = 1.

3. Show that, log 6 5 . log 7 6 . log 8 7 . log5 8 = 1.

4. Show that, log 8 9 . log 7 8 . log 9 7 = 1.

Have your essay responses graded by your teacher and enter your score here!

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