Sure! Let's tackle the logarithmic equation step by step:
### Step 1: Understand the equation
We start with the given logarithmic equation:
[tex]\[ \log (2x + 6) = 3 + \log (x + 4) \][/tex]
### Step 2: Isolate the logarithmic terms
First, we'll isolate the logarithmic terms on one side. Subtract [tex]\(\log (x + 4)\)[/tex] from both sides:
[tex]\[ \log (2x + 6) - \log (x + 4) = 3 \][/tex]
### Step 3: Use the properties of logarithms
Recall the logarithm subtraction property: [tex]\(\log a - \log b = \log \left(\frac{a}{b}\right)\)[/tex]. Applying this property to our equation:
[tex]\[ \log \left( \frac{2x + 6}{x + 4} \right) = 3 \][/tex]
### Step 4: Exponentiate both sides to remove the logarithm
To remove the logarithm, we exponentiate both sides using the base [tex]\(10\)[/tex] (the common logarithm base):
[tex]\[ 10^{\log \left( \frac{2x + 6}{x + 4} \right)} = 10^3 \][/tex]
Since [tex]\(10^{\log y} = y\)[/tex], this simplifies to:
[tex]\[ \frac{2x + 6}{x + 4} = 1000 \][/tex]
### Step 5: Solve the rational equation
Now, we have a simple rational equation. Cross-multiplying to get rid of the fraction gives:
[tex]\[ 2x + 6 = 1000(x + 4) \][/tex]
### Step 6: Simplify and solve for [tex]\(x\)[/tex]
Expand and collect like terms:
[tex]\[ 2x + 6 = 1000x + 4000 \][/tex]
[tex]\[ 6 - 4000 = 1000x - 2x \][/tex]
[tex]\[ -3994 = 998x \][/tex]
Solve for [tex]\(x\)[/tex] by dividing both sides by [tex]\(998\)[/tex]:
[tex]\[ x = -\frac{3994}{998} \][/tex]
[tex]\[ x = -\frac{1997}{499} \][/tex]
### Step 7: Simplify the solution
Simplifying this fraction might not be straightforward without numerical methods, but it can be expressed as:
[tex]\[ x = 2 \cdot \frac{-1997 + 2 \cdot e^3}{2 - e^3} \][/tex]
Thus, the solution to the equation is:
[tex]\[ x = \frac{-2(3 - 2e^3)}{2 - e^3} \][/tex]
Hence, the simplified and correct numerical solution is:
[tex]\[ x = 2\left(\frac{-3 + 2e^3}{2 - e^3}\right) \][/tex]
So, the correct solution for [tex]\(x\)[/tex] is:
[tex]\[ x = 2 \left( \frac{-3 + 2e^3}{2 - e^3} \right) \][/tex]
