Final answer:
The equation log(81–x) - log x = 2 simplifies to log((81–x)/x) = 2 and then to (81-x)/x = 100. Solving for x gives 81 = 101x, leading to x = 81/101, which does not match the provided options; thus x has no real solution from the given choices.
Step-by-step explanation:
The original equation given is log(81–x) - log x = 2. This equation can be simplified using the property of logarithms stating that the logarithm of a quotient is equal to the difference of the logarithms of the numerator and the denominator. Therefore, we can combine the two logarithms into a single logarithmic expression: log((81–x)/x) = 2.
To find the value of x, we can rewrite the logarithmic equation into its exponential form. Since log base 10 of a number is equal to 2, this means that the number is 10^2, which is 100. Thus, we get the equation (81-x)/x = 100.
To solve for x, we multiply both sides by x to get 81 - x = 100x and then add x to both sides to obtain 81 = 101x. Finally, we divide both sides by 101 to find x = 81/101, which simplifies down to approximately x = 0.80198, representing no exact match to the options given in the problem. Therefore, the correct answer would be that x has no real solution that matches the provided options.