What is the solution of [tex]log _x 729=3[/tex]?
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 logarithmic equation lo g x 729 = 3 . Our goal is to find the value of x that satisfies this equation.
Converting to Exponential Form To solve for x , we need to rewrite the logarithmic equation in its equivalent exponential form. Recall that lo g b a = c is equivalent to b c = a . Applying this to our equation, we have x 3 = 729 .
Finding the Cube Root Now we need to find the cube root of 729 to find the value of x . That is, we need to find a number x such that when it is multiplied by itself three times, the result is 729. We can write this as x = 3 729 .
Calculating the Cube Root We can find the cube root of 729 by trial and error or by recognizing that 9 3 = 9 × 9 × 9 = 81 × 9 = 729 . Therefore, x = 3 729 = 9 .
Final Answer Thus, 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 on the Richter scale, determining the pH of a solution in chemistry, and calculating the loudness of sound in decibels. In computer science, logarithms are used in the analysis of algorithms to measure their efficiency. For example, the time complexity of binary search is logarithmic, meaning the time it takes to search a sorted array grows logarithmically with the size of the array. Understanding logarithms helps us to solve problems related to exponential growth and decay, which are common in various scientific and engineering fields. Logarithmic scales are also used in finance to analyze investment growth and risk.
The solution to the logarithmic equation lo g x 729 = 3 is found by converting it to exponential form as x 3 = 729 . Solving for x gives us x = 9 . Thus, the final answer is x = 9 .
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