log6(x+7)−log6x=2

Solve for x

1/5

To solve the equation log6(x+7) - log6x = 2, we can use logarithmic properties to simplify the equation and find the value of x.

Step 1: Apply the quotient rule of logarithms. According to the quotient rule, the logarithm of the quotient of two numbers is equal to the logarithm of the numerator minus the logarithm of the denominator.

log6(x+7) - log6x = log6((x+7)/x) = 2

Step 2: Rewrite the equation using exponentiation. Since the log6 and the 6 on the right side of the equation cancel each other out, we can rewrite the equation as:

(x+7)/x = 6^2

Simplifying further:

(x+7)/x = 36

Step 3: Cross-multiply the equation:

36x = x + 7

Step 4: Solve for x by bringing all terms involving x to one side of the equation:

36x - x = 7

35x = 7

Step 5: Divide both sides of the equation by 35 to isolate x:

x = 7/35

Step 6: Simplify the fraction:

x = 1/5

Therefore, the solution to the equation log6(x+7) - log6x = 2 is x = 1/5.