Question

Given logbM=3.2 logbN=-1.5 and logbP=2.4 find the value of logb(m^2N/p)

Answer 1

The formulas:
$\log_a (xy)=\log_a x + \log_a y \\ \log_a (\frac{x}{y})=\log_a x - \log_a y \\ \log_a (x^n)n \log_a x$

Expand the expression:
$\log_b (\frac{m^2n}{p})=\log_b m^2 + \log_b n - \log_b p= 2 \log_b m+\log_b n - \log_b p$

Plug the values into the expression:
$\log_b m=3.2 \\ \log_b n=-1.5 \\ \log_b p=2.4 \\ \\ 2 \log_b m+\log_b n - \log_b p=2 \times 3.2-1.5-2.4=6.4-3.9=2.5 \\ \\ \boxed{\log_b (\frac{m^2n}{p})=2.5}$