Solve for x

Problem

ln(x+3)=ln(x)+ln(3)

Solution

1. Identify the domain of the equation by ensuring the arguments of the logarithms are positive, which requires x+3>0 and x>0 resulting in x>0

2. Apply the product rule for logarithms to the right side of the equation, where ln(a)+ln(b)=ln(a*b)

ln(x)+ln(3)=ln(3*x)

3. Equate the arguments of the logarithms since the natural log function is one-to-one.

ln(x+3)=ln(3*x)

x+3=3*x

4. Isolate the variable x by subtracting x from both sides of the equation.

3=2*x

5. Solve for x by dividing both sides by 2.

x=3/2

6. Verify that the solution x=3/2 is within the domain x>0

Final Answer

ln(x+3)=ln(x)+ln(3)⇒x=3/2

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