What is the solution of [tex]log _x 729=3[/tex]?
A. 6
B. 9
C. 18
D. 81
Answer (2)
Rewrite the logarithmic equation in exponential form: x 3 = 729 .
Take the cube root of both sides: x = 3 729 .
Calculate the cube root: x = 9 .
The solution is 9 .
Explanation
Understanding the Problem We are given the equation lo g x 729 = 3 . This means that x raised to the power of 3 equals 729. In other words, we need to find a number x such that x 3 = 729 .
Finding the Cube Root To find the value of x , we need to take the cube root of 729. That is, we need to calculate 3 729 .
Calculating the Cube Root We can find the cube root of 729 by trial and error or by using a calculator. We are looking for a number that, when multiplied by itself three times, equals 729. We know that 1 0 3 = 1000 , so the number we are looking for is less than 10. Let's try 9: 9 × 9 × 9 = 81 × 9 = 729 . So, 3 729 = 9 .
Final Answer Therefore, the solution to the equation lo g x 729 = 3 is x = 9 .
Examples
Logarithms are used in many real-world applications, such as measuring the intensity of earthquakes (the Richter scale), measuring the loudness of sound (decibels), and in calculating the pH of a solution in chemistry. In computer science, logarithms are used in the analysis of algorithms to measure their efficiency. For example, the time it takes to search for an item in a sorted list using binary search is proportional to the logarithm of the number of items in the list. The equation lo g x 729 = 3 is a specific example of how logarithms can be used to solve for an unknown base, which can be useful in various scientific and engineering contexts.
The solution to the equation lo g x 729 = 3 is x = 9 . This was found by converting the logarithmic equation to exponential form and then calculating the cube root of 729. Therefore, the answer is option B: 9.
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