Solve the logarithmic equation.
$\log\left(x+2\right)+\log2=3$

• A. 498
• B. 1998
• C. 6
• D. No Solution

Answer: (A)

Verified step by step guidance

1. Start by using the properties of logarithms to combine the logarithmic terms. Recall that \( \log(a) + \log(b) = \log(ab) \). Apply this to the equation: \( \log(x+2) + \log(2) = \log(2(x+2)) \).
2. Now, rewrite the equation using the combined logarithm: \( \log(2(x+2)) = 3 \).
3. To eliminate the logarithm, rewrite the equation in exponential form. Recall that if \( \log_b(a) = c \), then \( a = b^c \). Here, the base is 10 (common logarithm), so \( 2(x+2) = 10^3 \).
4. Calculate \( 10^3 \) to find the value on the right side of the equation: \( 10^3 = 1000 \). Thus, the equation becomes \( 2(x+2) = 1000 \).
5. Solve for \( x \) by first dividing both sides by 2: \( x+2 = 500 \). Then, subtract 2 from both sides to isolate \( x \): \( x = 498 \).