Questions: Solve for (x). [ log (x+6)-log 2=log x ] [ x= ]

Transcript text: Solve for $x$. \[ \log (x+6)-\log 2=\log x \] \[ x= \]

Solution

Step 1: Apply Logarithmic Properties

The given equation is:

\[ \log (x+6) - \log 2 = \log x \]

We can use the logarithmic property $\log a - \log b = \log \left(\frac{a}{b}\right)$ to combine the logarithms on the left side:

\[ \log \left(\frac{x+6}{2}\right) = \log x \]

Step 2: Equate the Arguments

Since the logarithms are equal, we can equate their arguments:

\[ \frac{x+6}{2} = x \]

Step 3: Solve for $x$

To solve for $x$, first clear the fraction by multiplying both sides by 2:

\[ x + 6 = 2x \]

Subtract $x$ from both sides:

\[ 6 = x \]

The solution to the equation is:

\[ \boxed{x = 6} \]

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