Find the logarithm. [tex]$\log _2 16$[/tex]
Answer (1)
We are asked to find lo g 2 16 .
We set x = lo g 2 16 , which means 2 x = 16 .
We express 16 as a power of 2: 16 = 2 4 .
Therefore, x = 4 , so lo g 2 16 = 4 .
Explanation
Understanding the problem We are asked to find the logarithm of 16 with base 2, which is written as lo g 2 16 . In simpler terms, we need to find the power to which we must raise 2 to get 16.
Setting up the equation Let's set x = lo g 2 16 . This means we are looking for x such that 2 x = 16 .
Expressing 16 as a power of 2 We know that 16 can be expressed as 2 4 since 2 × 2 × 2 × 2 = 16 . So, we can rewrite the equation as 2 x = 2 4 .
Equating the exponents Since the bases are the same (both are 2), we can equate the exponents. Therefore, x = 4 .
Finding the logarithm So, lo g 2 16 = 4 . This means that 2 raised to the power of 4 equals 16.
Examples
Logarithms are incredibly useful in many real-world situations. For example, they are used to measure the magnitude of earthquakes on the Richter scale. They also appear in calculations involving compound interest, population growth, and radioactive decay. Understanding logarithms helps us to work with very large or very small numbers in a more manageable way.
