Logarithms are the inverse of exponentials. If b to the power of y equals x, then the logarithm base b of x equals y. For example, since 2 squared equals 4, the log base 2 of 4 is 2. Today, we'll explore how to rewrite logarithms using natural logarithms. Our specific problem is to rewrite log base 2 of 16 using natural logarithms.
To convert between logarithm bases, we use the change of base formula. The formula states that the logarithm base a of x equals the natural logarithm of x divided by the natural logarithm of a. This works because of the fundamental properties of logarithms. We can actually use any base in the denominator, but natural logarithms (base e) are commonly used in mathematics. This formula allows us to compute logarithms in any base using calculators that only have natural logarithm functions.
Now let's apply the change of base formula to our problem: log base 2 of 16. First, we substitute into the formula to get natural log of 16 divided by natural log of 2. Next, we recognize that 16 is 2 to the 4th power, so the natural log of 16 is the natural log of 2 to the 4th power. Using the logarithm property that the log of a power equals the exponent times the log of the base, this becomes 4 times the natural log of 2. Finally, when we divide 4 times natural log of 2 by natural log of 2, the natural log of 2 terms cancel out, leaving us with 4. Therefore, log base 2 of 16 equals 4.
Let's verify our answer and explore alternative approaches. First, we can verify that our answer is correct by checking that 2 to the 4th power equals 16, which it does. Therefore, the log base 2 of 16 is indeed 4. As an alternative approach, we can directly apply the logarithm power rule to the numerator of our change of base formula. The natural log of 16 is the natural log of 2 to the 4th power, which equals 4 times the natural log of 2. When we substitute this into our formula, we get 4 times natural log of 2 divided by natural log of 2, which simplifies to 4. Both methods confirm that our answer is correct.
Let's work through more examples to solidify our understanding. First, let's find the log base 2 of 8 using natural logarithms. We apply the change of base formula to get natural log of 8 divided by natural log of 2. Since 8 is 2 cubed, the natural log of 8 is 3 times the natural log of 2. Dividing by natural log of 2 gives us 3. Next, let's find the log base 2 of 32. Using the same approach, we get natural log of 32 divided by natural log of 2. Since 32 is 2 to the 5th power, this simplifies to 5 times natural log of 2 divided by natural log of 2, which equals 5. These examples show a pattern: when working with powers of 2, the logarithm base 2 is simply the exponent. This pattern helps us quickly verify our answers when using the change of base formula.