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

["Solving ( \log_b(64) = 3 ): Finding the Base ( b )", "Understanding logarithms is essential in math, and one common question involves solving equations of the form ( \log_b(x) = y ). This article explains how to find the base ( b ) given the equation ( \log_b(64) = 3 ), using the definition of logarithms and clear, step-by-step reasoning.", "### What Does ( \log_b(64) = 3 ) Mean?", "The logarithmic expression ( \log_b(64) = 3 ) asks: To what power must the base ( b ) be raised to equal 64? In equation form, this means:", "[\nb^3 = 64\n]", "We now solve for ( b ) to determine its value.", "### Step-by-Step Solution", "Step 1: Express 64 as a Power of 2\nTo simplify, rewrite 64 as a power of 2:\n[\n64 = 2^6\n]", "Step 2: Substitute into the Equation\nUsing the substitution, the equation becomes:\n[\nb^3 = 2^6\n]", "Step 3: Solve for ( b )\nTo isolate ( b ), take the cube root of both sides:\n[\nb = \sqrt[3]{2^6}\n]", "Using the exponent rule ( \sqrt[n]{a^m} = a^{m/n} ), this simplifies to:\n[\nb = 2^{6/3} = 2^2\n]", "Step 4: Compute the Final Value\nNow simplify:\n[\nb = 4\n]", "---", "### Verification\nCheck that ( \log_4(64) = 3 ):\nSince ( 4^3 = 64 ), it’s correct.", "---", "### Summary", "Given ( \log_b(64) = 3 ), we found:", "[\nb = 4\n]", "This result connects logarithmic identities with exponent rules, showing how different bases relate to powers of numbers. Whether you’re solving equations or exploring logarithmic scales, mastering how to extract bases like ( b ) from ( \log_b(x) = y ) is key to deeper mathematical fluency.", "---", "Keywords: ( \log_b(64) = 3 ), solve for ( b ), logarithmic equation, base ( b ), exponential forms, math tip, logarithms explained.\nMeta Description: Solve ( \log_b(64) = 3 ) by rewriting 64 as a power of 2, then solving ( b^3 = 64 ). Find ( b = 4 ) using exponent rules.\nTopics: Logarithms, solving logarithmic equations, finding unknown bases, exponent roots, math fundamentals."]