Fill in the blank: [tex]$\log _3 243=$[/tex] ______
Answer (2)
We are asked to evaluate the logarithm lo g 3 243 .
We need to find the exponent x such that 3 x = 243 .
We find that 3 5 = 243 .
Therefore, lo g 3 243 = 5 .
Explanation
Understanding the problem We are asked to evaluate the logarithm lo g 3 243 . This means we need to find the exponent to which we must raise 3 to obtain 243.
Expressing 243 as a power of 3 We need to express 243 as a power of 3. Let's find x such that 3 x = 243 . We can calculate powers of 3 until we reach 243:
3 1 = 3 3 2 = 9 3 3 = 27 3 4 = 81 3 5 = 243
Finding the logarithm Since 3 5 = 243 , we have lo g 3 243 = 5 .
Final Answer Therefore, the answer is 5.
Examples
Logarithms are used in many real-world applications, such as measuring the magnitude of earthquakes on the Richter scale, determining the acidity or alkalinity (pH) of a solution, and modeling population growth. For example, the Richter scale uses logarithms to quantify the size of an earthquake, where each whole number increase represents a tenfold increase in amplitude. Understanding logarithms helps scientists and engineers analyze and interpret data in these fields.
The value of lo g 3 243 is 5, as 3 raised to the power of 5 equals 243. This demonstrates the relationship between logarithms and exponents.
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