Write in Exponential Form log of x+14- log of x = log of x+6

Problem

(log_)(x+14)−(log_)(x)=(log_)(x+6)

Solution

1. Apply the quotient rule for logarithms to the left side of the equation, which states that (log_)(a)−(log_)(b)=(log_)(a/b)

(log_)((x+14)/x)=(log_)(x+6)

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

(x+14)/x=x+6

3. Isolate the numerator by multiplying both sides of the equation by x

x+14=x*(x+6)

4. Distribute the x on the right side to prepare the equation for standard form.

x+14=x2+6*x

5. Rearrange into standard quadratic form by subtracting x and 14 from both sides.

0=x2+5*x−14

6. Convert to exponential form by recognizing that the logarithmic equation (log_b)(y)=x is equivalent to bx=y In the original expression, the base is 10

10=(x+14)/x

Final Answer

(log_)(x+14)−(log_)(x)=(log_)(x+6)⇒(x+14)/x=x+6

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