Final answer:
To solve the equation (log3x)² – 12 log3x², simplify the equation using logarithmic properties and properties of exponents. The equation can be rewritten as log3(1/x^10), which means that log3(1/x^10) = 0. Using the property that if logb(x) = y, then b^y = x, we can solve for x by equating 3^0 to 1/x^10. However, this leads to a contradiction, indicating that there is no solution for x in this equation.
Step-by-step explanation:
To solve the equation (log3x)² – 12 log3x², we can apply logarithmic properties.
First, let's simplify the equation. Using the property that (logab)2 = logab2, we can rewrite the equation as log3x2 - 12 log3x². Using the property that logab - logac = loga(b/c), we can simplify further to log3(x2 / x²12). This can be reduced to log3(1 / x¹⁰) or log31 - log3x10.
Since log31 is equal to 0, the equation becomes -log3x10. We can rewrite this as log31/x10 using the property that -log3a = log3(1/a). Thus, the equation simplifies to log31/x10.
Now, we can solve for x. By applying the property that if logbx = y, then by = x, we have 3log31/x10 = 1/x10. This can be further simplified as (3/x)10 = 1/x10. Taking the 10th root of both sides, we get 3/x = 1/x. Multiplying both sides by x, we have 3 = 1. However, since 3 is not equal to 1, there is no solution for x in this equation.