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.

a. log₃(x+2)+log₃x-log₃2

b. 2log_am-3log_an

c. ½log_bm+3/2log_b2n-log_bm²n(Be sure to simplify, only positive, whole number exponents. Radical symbol over numerator and denominator is acceptable).

Solution

a) log3(x+2)+log3x-log32 = log[3(x+2)*3x/32] = log[9x(x + 2)/32]

b)  2logam-3logan = log(am)^2 - log(an)^3 = log[(am)^2/(an)^3] = log(m²/an³)

c) ½logbm+3/2logb2n-logbm2n = log(bm)^1/2 + log(2n)^3/2 - logbm2n = log[(bm)^1/2*(2n)^3/2/bm2n ] = log[(2n/bm)^1/2]