Question

GIven that log₃6=m, and log₆5=n express log₃10 in terms of m and n

Answer 1

Use the formula $log_a (bc)=log_a b +log_a c$:

$$log_3 10 =log_3 (2\cdot 5)=log_3 2+log_3 5$$

Express $log_3 2$ from the first equality:

$$log_3 6 = m$$ $$log_3 (2\cdot3)=m$$ $$log_3 2+log_3 3=m$$ $$log_3 2+1=m$$ $$log_3 2=m-1$$

Express $log_3 5$ by using formulas $log_a b=\frac{1}{log_b a}, \frac{log_a b}{log_a c}=log_c b$

$$log_3 5=\frac{log_6 5}{log_6 3}=log_6 5\cdot log_3 6=nm$$

Thus,

$$log_3 10 =log_3 2+log_3 5 =m-1+nm=mn+m-1$$

Answer: $mn+m-1$