Using the laws of logarithms please evaluate the following questions. Make sure to leave answers in exact form.

Question

a) log₃18 + log₃5 - log₃10

b) 3log₂(1/2) - 2log₂(1/4)

Answer 1

a log₃9 = 2 exactly

add logs by multiplying, and subtract logs by dividing

log18 + log5 - log 10 = log(18)(5)/10 = log90/10 = log9 log to base 3 of 9 = the exponent you need to put over 3 to get 9 3^2 =9 log₃9 = exactly 2

log₃18 = ln18/ln3 = 2.89/1.099 = about 2.63

log₃ 5= ln5/ln3 = 1.61/1.099 = 1.46

log₃10 = ln10/ln3 = 2.10

2.63+1.49-2.1 =with rounding errors still close to 2

b 3log(1/2) -2log(1/4) = 3(-1) -2(-2) = -3 +4 = exactly 1

log₂(1/2) = -1, because 2^-1 = 1/2

log₂(1/4) = -2 as 2^-2 = 1/2^2 = 1/4

Answer 2

a) log₃18 + log₃5 - log₃10 = log₃(18·5/10) = log₃9 = 2

b) 3log₂(1/2) - 2log₂(1/4) = 3(- 1) - 2(-2) = - 3 + 4 = 1