What is the value of $\log _{27} 9$?
Answer (1)
Let x = lo g 27 9 .
Rewrite in exponential form: 2 7 x = 9 .
Express both sides as powers of 3: ( 3 3 ) x = 3 2 ⇒ 3 3 x = 3 2 .
Equate exponents and solve for x : 3 x = 2 ⇒ x = 3 2 . The final answer is 3 2 .
Explanation
Understanding the Problem We are asked to find the value of lo g 27 9 . This means we need to find the exponent to which we must raise 27 to get 9.
Converting to Exponential Form Let x = lo g 27 9 . We can rewrite this equation in exponential form as 2 7 x = 9 .
Expressing as Powers of 3 Now, we express both sides of the equation as powers of 3. Since 27 = 3 3 and 9 = 3 2 , we have ( 3 3 ) x = 3 2 .
Simplifying the Equation Using the power of a power rule, we simplify the left side to get 3 3 x = 3 2 .
Equating Exponents Since the bases are equal, we can equate the exponents: 3 x = 2 .
Solving for x Finally, we solve for x by dividing both sides by 3: x = 3 2 . Therefore, lo g 27 9 = 3 2 .
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 to analyze the efficiency of algorithms. 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. Understanding logarithms helps in these diverse fields to quantify and analyze phenomena that vary over a wide range of values.
