How do you simplify log_20 (8000^x)?

1 Answer
EZ as pi
Sep 11, 2016

3x

Explanation:

Recall:

The " power law " of logs states:

log_p q^color(red)(m) = color(red)(m)log_p q

The " change of base law " states

log_a b = (log_c b)/(log_c a)

("We usually use " log_10" as " log_c)

Apply these two laws to the question:

log_20 8000^color(red)(x) = color(red)(x)log_20 8000

=color(red)(x)((log_10 8000)/(log_10 20)) " "larr use a calculator

=x xx 3 = 3x

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
However, using another approach - the maths is beautiful, and we do not even need a calculator this time!

Note that 20^3 = 8000

log_20 8000^x = log_20 (20^3)^x

=log_20 20^(3x)

=3x log_20 20 " "larr log_20 20 = 1

3x

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