Solve for x

Problem

(log_9)(x+6)−(log_9)(x)=(log_9)(2)

Solution

1. Apply the quotient rule for logarithms, which states that (log_b)(M)−(log_b)(N)=(log_b)(M/N)

(log_9)((x+6)/x)=(log_9)(2)

2. Use the property of equality for logarithms, which states that if (log_b)(M)=(log_b)(N) then M=N

(x+6)/x=2

3. Multiply both sides by x to eliminate the fraction.

x+6=2*x

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

6=x

5. Verify the solution by checking if the arguments of the original logarithms are positive. Since 6+6>0 and 6>0 the solution is valid.

x=6

Final Answer

(log_9)(x+6)−(log_9)(x)=(log_9)(2)⇒x=6

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