If \( \log_2(x) + \log_2(8) = 5 \), find \( x \).
["Solving ( \log_2(x) + \log_2(8) = 5 ): A Complete Step-by-Step Guide", "If you're working with logarithmic equations like ( \log_2(x) + \log_2(8) = 5 ), it's essential to understand the fundamental properties of logarithms to solve for ( x ) efficiently. This article guides you through solving this equation using step-by-step algebraic techniques, while also optimizing for search engines with clear keywords, structured content, and useful mathematical explanation.", "---", "### Understanding the Equation", "We begin with the equation:", "[\n\log_2(x) + \log_2(8) = 5\n]", "This equation combines two logarithmic expressions with the same base, 2. The key logarithmic property we’ll use is:", "[\n\log_b(a) + \log_b(c) = \log_b(ac)\n]", "This allows us to combine the two logs on the left into a single logarithmic expression.", "---", "### Step 1: Combine Logs Using Log Multiplication Rule", "Apply the product rule:", "[\n\log_2(x) + \log_2(8) = \log_2(8x)\n]", "So the equation becomes:", "[\n\log_2(8x) = 5\n]", "---", "### Step 2: Convert Logarithmic to Exponential Form", "To eliminate the logarithm, rewrite in exponential form using the definition of logarithms:", "[\n\log_b(A) = C \quad \Leftrightarrow \quad b^C = A\n]", "Thus,", "[\n8x = 2^5\n]", "Compute ( 2^5 = 32 ):", "[\n8x = 32\n]", "---", "### Step 3: Solve for ( x )", "Divide both sides by 8:", "[\nx = \frac{32}{8} = 4\n]", "---", "### Step 4: Verify the Solution", "Plug ( x = 4 ) back into the original equation to confirm:", "[\n\log_2(4) + \log_2(8) = \log_2(4 \cdot 8) = \log_2(32)\n]", "Since ( 2^5 = 32 ), we know ( \log_2(32) = 5 ), so the equation holds true.", "---", "### Final Answer", "[\n\boxed{x = 4}\n]", "---", "### SEO Optimization & Keywords", "To maximize visibility for users searching for logarithmic equation solutions like this, the article integrates the following SEO-friendly elements:", "- Primary keywords: "solve ( \log_2(x) + \log_2(8) = 5 )", "how to solve logarithmic equations", "log base 2 logarithmic equation".\n- Clear structure: Headings, steps, and summaries improve readability and support search algorithms.\n- Explanation-based content: Focuses on logic and methodology, not just the answer.\n- Practical verification: Includes back-substitution to build trust and accuracy.\n- Long-tail integration: These themes often match phrases like "step-by-step solution logarithms base 2" frequently searched by students and learners.", "---", "### Summary", "Solving ( \log_2(x) + \log_2(8) = 5 ) involves combining logs, converting to exponential form, and solving algebraically. The correct solution is ( x = 4 ). This guide offers clear insight for learners studying logarithmic properties and practical problem-solving.", "For further reading, explore:\n- Logarithmic identities and their applications\n- Solving equations involving logarithmic functions of different bases\n- Step-by-step guide to exponential equations", "---", "Keywords: ( \log_2(x) + \log_2(8) = 5 ), solve logarithmic equation, step-by-step logarithms, math problem solution, logarithm basics, ( x = 4 )", "---", "Optimizing both content and clarity helps users not only find the answer but also understand the underlying math—making this article a valuable SEO resource for students, teachers, and math enthusiasts."]
