If \(\log_b(64) = 3\), find \(b\).

["How to Solve For (b) in the Equation (\log_b(64) = 3): A Step-by-Step Guide", "Understanding logarithmic equations is essential in mathematics, especially when solving for bases like (b). One commonly encountered problem is:", "[\n\log_b(64) = 3\n]", "If you’re wondering “What is the value of (b)?”, this article will walk you through the step-by-step solution using the definition of logarithms.", "---", "### What Does (\log_b(64) = 3) Mean?", "By definition, the logarithmic equation (\log_b(a) = c) means that (b^c = a). Applying this to our equation:", "[\n\log_b(64) = 3 \quad \Rightarrow \quad b^3 = 64\n]", "So, solving for (b) reduces to finding the base (b) such that when raised to the power of 3 equals 64.", "---", "### Step 1: Express 64 as a Power of 2", "We know that 64 is a power of 2:", "[\n64 = 2^6\n]", "Rewriting our equation:", "[\nb^3 = 2^6\n]", "---", "### Step 2: Solve for (b)", "Take the cube root of both sides:", "[\nb = \sqrt[3]{2^6}\n]", "Using exponent rules:", "[\nb = 2^{6/3} = 2^2 = 4\n]", "---", "### Step 3: Verify the Solution", "Check that (\log_4(64) = 3):", "[\n4^3 = (2^2)^3 = 2^6 = 64\n]", "Since this is true, our solution is correct.", "---", "### Conclusion", "If (\log_b(64) = 3), then the base (b) must be:", "[\n\boxed{4}\n]", "This problem illustrates the core concept that logarithms convert exponentiation into linear form, allowing easy extraction of the base.", "---", "Keywords:\n- (\log_b(64) = 3) solution\n- How to find base from logarithm\n- Solve (\log_b(x) = c)\n- Mathematical explanation (\log_b(a) = c)\n- Step-by-step logarithmic equation\n- Find (b) from (\log_b(64) = 3)", "Meta Description:\nSolve (\log_b(64) = 3) and find the base (b). Step-by-step explanation with verification and logarithmic principles for students and math enthusiasts."]