Write each expression as a single logarithm with coefficient

Write each expression as a single logarithm with coefficient 1. Assume all variables represent positive real numbers, with a ? 1 and b ? 1. Show your work and identify the logarithm property used.

Solution

a. log₃(x+2) + log₃x- log₃2

We have to use the following log rules

log_ab+log_ac=log_abc

log_ab-log_ac=log_a(b/c)

log₃(x+2) + log₃x - log₃2=log₃(x(x+2))-log₃2=log(x(x+2)/3)

b. 2log_am -3log_an

We have to use the following log rule here

a log_bc=log_bc^a

2log_am-3log_an= log_am²-log_an³ =log_a(m²/n³)

c. 1/2log_bm +3/2log_b2n -log_bm²n

log_bm^1/2 + log_b(2n)^3/2-log_bm²n

log_b((m^1/2 * 8^1/2n^3/2)/m²n)

log_b(8^1/2n^1/2/(m^3/2))