Final answer:
To solve the equation log(x)((1)/(125))=-(3)/(2), rewrite it using logarithmic properties and set the resulting exponents equal to each other. The exact solution is x = (1)/(2).
Step-by-step explanation:
To solve the equation logx((1)/(125))=-(3)/(2), we can rewrite it using the logarithmic property which states that loga(b) = c is equivalent to ac = b. Applying this property, we get x-(3)/(2) = (1)/(125). Next, we can rewrite the right side of the equation as a power of x, so we have x-(3)/(2) = x-3. Setting the exponents equal to each other, we obtain -(3)/(2) = -3. To solve for x, we multiply both sides of the equation by -2, which gives us 3 = 6x. Finally, we divide both sides by 6 to find the exact solution: x = (1)/(2).
To solve the equation logx(1/125) = -3/2, we can rewrite it using the definition of a logarithm as an exponent. The equation states that x raised to the power of -3/2 equals 1/125. Rewriting 1/125 as its prime factorization gives us 1/53. Since 5 to the power of 3 is 125, we have x-3/2 equals 5-3. Therefore, x must be equal to 5 because both the base and the exponent must match for the equation to hold true.