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Question

Which equation is equivalent to log3(2x4 + 8x3) – 3log3x = 2log3x?

Answer

The solution is log3(2x+8) = log3 x². Step-by-step explanation: Step 1: Simplify the given equation by using the properties of logarithms. log3(2x^4+8x^3) - 3log3x = 2log3x Step 2: Simplify the left-hand side of the equation by combining the logs. log3(2x^4+8x^3) = 2log3x + 3log3x Step 3: Simplify the right-hand side of the equation by converting the logs to base 2. log2(2x^4+8x^3) = 5log2x Step 4: Rewrite the equation in terms of log3. log3(x³) + log3(2x+8) = 5log3x Step 5: Simplify the equation by combining the logs and using the properties of logarithms. 3log3x + log3(2x+8) = 5log3x log3(2x+8) = 2log3x Step 6: Rewrite the equation in exponential form. 3^2 = 2x + 8 x^2 = (2x+8) / 3 Step 7: Simplify the equation. 3x^2 = 2x + 8 x^2 - 2x - 8 = 0 (x-4)(x+2) = 0 Therefore, x = 4 or x = -2. Step 8: Check for extraneous solutions. Since x = -2 is not a valid solution (logarithms of negative numbers are undefined), the only valid solution is x = 4. Step 9: Substitute the solution into the original equation to verify. log3(2x+8) = log3(2*4+8) = log3(16) log3 x² = log3 4² = log3 16 The solutions match, so the answer is verified. Therefore, the final answer is log3(2x+8) = log3 x².