How to simplify logarithm?

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Can anyone please explain to me how to do question d, when there I see different bases? Thanks!

1 Answer
Sep 23, 2017

log_4 32-log_9 27=1

Explanation:

"there are 2 possible approaches"

(1)color(blue)" Evaluation"

•color(white)(x)log_b x=nhArrx=b^n

rArrlog_4 32=nrArr32=4^n

"express the equation in base 2"

rArr2^5=(2)^(2n)

"equating the exponents"

2n=5rArrn=5/2color(red)(larr)

"similarly for "log_9 27=n

rArr27=9^n

"expressing the equation in base 3"

3^3=3^(2n)

rArr2n=3rArrn=3/2color(red)(larr)

rArrlog_4 32-log_9 27=5/2-3/2=1

(2)color(blue)" Change of base"

•color(white)(x)log_b a=log_c a/log_c b

"choose base 10 as this can be evaluated on calculator"

rArrlog_4 32-log_9 27

=log_10 32/log_10 4-log_10 27/log_10 9

=2.5-1.5=1