Solve the Equation: 2log(x+4) = 1 | Step-by-Step Solution

To solve the equation $2\log(x+4) = 1$, we follow these steps:

• Step 1: Divide both sides by 2 to simplify the equation.
• Step 2: Apply the logarithm property to rewrite the equation.
• Step 3: Convert the logarithmic equation into an exponential equation.
• Step 4: Solve the resulting equation for $x$.

Let's work through the steps:

Step 1: Start by dividing both sides of the equation by 2:

$\log(x+4) = \frac{1}{2}$

Step 2: Translate the logarithmic equation to its exponential form. Recall that $\log_b(A) = C$ implies $b^C = A$. Here, the base is 10 (since it's a common logarithm when the base is not specified):

$x+4 = 10^{\frac{1}{2}}$

Step 3: Simplify $10^{\frac{1}{2}}$ which is the square root of 10:

$x+4 = \sqrt{10}$

Step 4: Solve for $x$ by isolating it:

$x = \sqrt{10} - 4$

Thus, the value of $x$ is $-4 + \sqrt{10}$.