What is the solution of $\log _x 728=3$?
Answer (2)
Rewrite the logarithmic equation lo g x 728 = 3 in exponential form as x 3 = 728 .
Take the cube root of both sides to get x = 3 728 .
Approximate the cube root of 728, which is close to 9 since 9 3 = 729 .
Check the given options (6, 0, 18, 81) to see if any satisfy the equation, but none do. Therefore, there may be an error in the question or options.
Explanation
Understanding the Problem We are given the equation lo g x 728 = 3 . Our goal is to find the value of x that satisfies this equation. This involves understanding the relationship between logarithms and exponents.
Converting to Exponential Form To solve for x , we need to rewrite the logarithmic equation in its equivalent exponential form. The general rule is that lo g b a = c is equivalent to b c = a . Applying this to our equation, we get x 3 = 728 .
Finding the Cube Root Now we need to find the cube root of 728 to solve for x . That is, we need to find a number x such that when it is multiplied by itself three times, it equals 728. So, x = 3 728 .
Approximating the Solution We can approximate the cube root of 728. Since 9 3 = 729 , we can deduce that 3 728 is very close to 9. Let's check if 9 is indeed the answer. If x = 9 , then x 3 = 9 3 = 729 . This is very close to 728, but not exactly 728. However, since the options provided are 6, 0, 18, and 81, and we know that the answer is close to 9, we should check if there's a typo in the problem. If the equation was lo g x 729 = 3 , then the solution would be exactly 9.
Analyzing the Options and Possible Errors Given the options, and the fact that 9 3 = 729 is very close to 728, it is likely that the problem intended to have 729 instead of 728. However, since we must choose from the given options, and none of them are close to the actual cube root of 728 (which is approximately 8.995), we must consider the possibility of an error in the question itself. If we assume that the question was meant to be lo g x 729 = 3 , then x 3 = 729 , and x = 3 729 = 9 . However, 9 is not among the options. The closest option to the real solution is none of the provided ones. However, if we assume that the problem intended to ask for lo g x 729 = 3 , then x = 9 . But 9 is not an option. Let's analyze the options:
If x = 6 , then 6 3 = 216 = 728 .
If x = 0 , then 0 3 = 0 = 728 .
If x = 18 , then 1 8 3 = 5832 = 728 .
If x = 81 , then 8 1 3 = 531441 = 728 .
Since none of the options satisfy the equation, and the closest integer solution (9) is not among the options, there might be an error in the question or the provided options. However, without additional information, we cannot definitively determine the correct answer. Given that 9 3 = 729 is the closest perfect cube to 728, it's possible the question intended to have 729 instead of 728. In that case, the answer would be 9, which is not among the options. Therefore, we cannot provide a definitive answer from the given choices.
Final Answer and Conclusion Since none of the provided options (6, 0, 18, 81) satisfy the equation lo g x 728 = 3 , and the closest integer solution (9) is not among the options, we can conclude that there might be an error in the question or the provided options. Therefore, we cannot provide a definitive answer from the given choices. However, if the question was lo g x 729 = 3 , then the answer would be 9.
Examples
Logarithms are used in many real-world applications, such as measuring the intensity of earthquakes on the Richter scale, determining the pH of a solution in chemistry, and calculating the loudness of sound in decibels. Understanding logarithms helps us to work with very large or very small numbers more easily. For example, the Richter scale uses logarithms to quantify the energy released by an earthquake, where each whole number increase represents a tenfold increase in amplitude. This allows scientists to represent a wide range of earthquake intensities in a manageable scale.
To solve lo g x 728 = 3 , we find that x 3 = 728 leads us to x = 3 728 , approximately 9. None of the available options (6, 0, 18, 81) satisfy this equation, indicating a potential error in the problem. Assuming a correction to 729 would yield a certain answer of 9, which is absent from the options.
;
