When you calculate (ln) 7, you would be finding the value of which of the following expressions?
The Correct Answer and Explanation is :
The correct expression for calculating (\ln(7)) is the natural logarithm of 7. Specifically, it refers to the logarithm of 7 to the base (e), where (e) is Euler’s number, approximately equal to 2.71828.
The Natural Logarithm:
The natural logarithm, denoted (\ln(x)), is the inverse operation of exponentiation with base (e). In other words, if (y = \ln(x)), then (e^y = x). The natural logarithm has many important applications in mathematics, physics, engineering, and economics, particularly in fields involving exponential growth or decay.
The Calculation of (\ln(7)):
To calculate (\ln(7)), you’re essentially solving for the power to which (e) must be raised to equal 7. In mathematical terms:
[
\ln(7) = y \quad \text{such that} \quad e^y = 7
]
This means you are looking for the exponent (y) such that when you raise (e) to this power, the result is 7. This value is not a simple whole number, but rather a decimal. Using a calculator or computational software, we find:
[
\ln(7) \approx 1.9459
]
Why the Natural Logarithm is Important:
The natural logarithm is useful because many natural processes, such as population growth, radioactive decay, and continuous compound interest, are modeled using exponential functions involving the base (e). For example, the formula for compound interest uses the natural logarithm to model how investments grow exponentially over time.
In summary, (\ln(7)) refers to the natural logarithm of 7, and is asking for the exponent (y) such that (e^y = 7). It is a commonly used mathematical operation in many areas of science and finance.