Solve for x

Problem

ln(x+9)=ln(x)+ln(9)

Solution

1. Apply the product rule for logarithms to the right side of the equation, which states that ln(a)+ln(b)=ln(a*b)

ln(x+9)=ln(9*x)

2. Exponentiate both sides using the base e to remove the natural logarithms, since the function ƒ(x)=ln(x) is one-to-one.

x+9=9*x

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

9=8*x

4. Solve for x by dividing both sides by 8

x=9/8

5. Verify the solution by ensuring that the arguments of all logarithms in the original equation are positive. Since 9/8>0 and 9/8+9>0 the solution is valid.

Final Answer

ln(x+9)=ln(x)+ln(9)⇒x=9/8

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