How to simplify logarithm?

enter image source here

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=1log432log927=1

Explanation:

"there are 2 possible approaches"there are 2 possible approaches

(1)color(blue)" Evaluation"(1) Evaluation

•color(white)(x)log_b x=nhArrx=b^nxlogbx=nx=bn

rArrlog_4 32=nrArr32=4^nlog432=n32=4n

"express the equation in base 2"express the equation in base 2

rArr2^5=(2)^(2n)25=(2)2n

"equating the exponents"equating the exponents

2n=5rArrn=5/2color(red)(larr)2n=5n=52

"similarly for "log_9 27=nsimilarly for log927=n

rArr27=9^n27=9n

"expressing the equation in base 3"expressing the equation in base 3

3^3=3^(2n)33=32n

rArr2n=3rArrn=3/2color(red)(larr)2n=3n=32

rArrlog_4 32-log_9 27=5/2-3/2=1log432log927=5232=1

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

•color(white)(x)log_b a=log_c a/log_c bxlogba=logcalogcb

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

rArrlog_4 32-log_9 27log432log927

=log_10 32/log_10 4-log_10 27/log_10 9=log1032log104log1027log109

=2.5-1.5=1=2.51.5=1