Write in Exponential Form log base 3 of x+5=2- log base 3 of x-3

Problem

(log_3)(x+5)=2−(log_3)(x−3)

Solution

1. Group the logarithmic terms by adding (log_3)(x−3) to both sides of the equation.

(log_3)(x+5)+(log_3)(x−3)=2

2. Apply the product rule for logarithms, which states that (log_b)(M)+(log_b)(N)=(log_b)(M*N)

(log_3)((x+5)*(x−3))=2

3. Convert to exponential form using the definition (log_b)(y)=x⇔bx=y

(x+5)*(x−3)=3

4. Simplify the exponent on the right side of the equation.

(x+5)*(x−3)=9

Final Answer

(log_3)(x+5)=2−(log_3)(x−3)⇒(x+5)*(x−3)=9

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