Logarithm problem 4 by Dhaval Furia

Neaten up Equation 1 using log properties: $\log_5(x+y)+\log_5(x-y)=3 \Longrightarrow \log_5(x^2-y^2)=\log_5(125) \Longrightarrow x^2-y^2=125$

Neaten up Equation 2 using log properties: $\log_2y-\log_2x=1-\log_23 \Longrightarrow \log_2\frac{y}{x}=\log_2\frac{2}{3} \Longrightarrow \frac{y}{x}=\frac{2}{3}$

Rearrange Equation 2: $\frac{y}{x}=\frac{2}{3} \Longrightarrow y=\frac{2}{3}x$

Substitute $x$ into Equation 1: $x^2-y^2=125 \Longrightarrow x^2-\left(\frac{2}{3}x\right)^2=125 \Longrightarrow x^2-\frac{4}{9}x^2=125 \Longrightarrow \frac{5}{9}x^2=125 \Longrightarrow x^2=225 \Longrightarrow \boxed{x=15}$

Find $y$ by substituting $x$ into Equation 2: $\frac{y}{x}=\frac{2}{3} \Longrightarrow \frac{y}{15}=\frac{2}{3} \Longrightarrow \boxed{y=10}$

Finally: $xy=15\times10=\boxed{150}$