Question

Solve the equation (log_{27}(log_x{10}) = frac{1}{3}).

a) (x = 27)
b) (x = 10)
c) (x = 3)
d) (x = 81)

Answer 1

To solve the equation \(log_{27}(log_x(10)) = \frac{1}{3}\), we rewrite the logarithms and equate the inner log to 3, solving for x gives us 103 or 1000, which is not one of the provided options. Hence, there might be a typo in the question as none of the options is a correct solution.

Step-by-step explanation:

The question involves solving the equation log27(logx(10)) = \(\frac{1}{3}\). To solve this, we first address the logarithm on the right side of the equation. The base of the outer log is 27, and since 27 is equal to 33, we can use the property that the log of a number raised to an exponent is the product of the exponent and the log of the number.

We have the following steps:

1. Recognize that log27(y) can be rewritten as log33(y) = \(\frac{1}{3}\)log3(y).
2. Thus, log27(logx(10)) = log33(logx(10)) which simplifies to \(\frac{1}{3}\)log3(logx(10)).
3. Since we are given that this expression is equal to \(\frac{1}{3}\), we can equate log3(logx(10)) to 1. This means that logx(10) must be equal to 3.
4. Looking at the inner log, we want to find an x such that when raised as a base of 10 will give us 3. The only number that satisfies this is 103 or 1000. However, this is not one of the options provided.
5. There might be a typo in the original equation or the answer choices. As such, none of the provided options (27, 10, 3, 81) is correct for the given equation.