Determine the value of log(1000).

To find the value of log(1000), we first need to clarify which logarithmic base we are using. The most common bases are 10 (common logarithm) and e (natural logarithm). For this explanation, I'll assume we're working with the common logarithm, which is typically denoted as log without any base specified.

Understanding Logarithms

A logarithm answers the question: "To what exponent must a base be raised to produce a given number?" In the case of log(1000), we are looking for the exponent to which 10 must be raised to yield 1000.

Breaking Down the Calculation

We can express 1000 as a power of 10:

• 1000 = 10 × 10 × 10 = 10^3

This means that 10 raised to the power of 3 equals 1000. Therefore, we can write:

log(1000) = log(10^3)

Applying Logarithmic Properties

One of the key properties of logarithms is that log(a^b) = b × log(a). Applying this property here:

log(10^3) = 3 × log(10)

Since log(10) equals 1 (because 10^1 = 10), we can simplify further:

log(1000) = 3 × 1 = 3

Final Result

Thus, the value of log(1000) is 3. This means that 10 raised to the power of 3 gives us 1000, confirming our calculation.

Real-World Application

Logarithms are widely used in various fields, including science, engineering, and finance. For example, they help in measuring sound intensity (decibels) and in calculating pH levels in chemistry. Understanding how to manipulate logarithms can be incredibly useful in these contexts.