How do you condense Log_4 (20) - Log_4 (45) + log_4 (144)?

2 Answers
May 27, 2018

3

Explanation:

"using the "color(blue)"laws of logarithms"

•color(white)(x)logx+logy=log(xy)

•color(white)(x)logx-logy=log(x/y)

•color(white)(x)log_b x=nhArrx=b^n

log_4 20+log_4 144-log_4 45

=log_4((20xx144)/45)

=log_4(64)=n

64=4^3=4^nrArrn=3

May 27, 2018

log_4(20/45*144) = log_4(64) = 3

Explanation:

The log product and quotient rules allow us to combine these terms, as long as the logs have the same base:
log_b(x) + log_b(y) = log_b(x*y)
log_b(x) - log_b(y) = log_b(x/y)

All the terms in this problem have logs of base 4, so we can apply these rules:
log_4(20) - log_4(45) + log_4(144)
= log_4(20/45*144)
= log_4(64)

Of course, the result of a log is the exponent of the base that will give you that number: log_b (x) = y means that b^y = x
So, since 4^3 = 64, then
log_4(64) = 3