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
Jim G.
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#

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