Find each value. If applicable, give an approximation to four decimal places. See Example 5. ln 0.00013

Verified step by step guidance

1. Recognize that the problem requires finding the natural logarithm of a number, specifically \( \ln(0.00013) \).
2. Recall that the natural logarithm function, \( \ln(x) \), is the inverse of the exponential function \( e^x \).
3. Use the property of logarithms that \( \ln(a \times b) = \ln(a) + \ln(b) \) to simplify the expression if needed. In this case, express 0.00013 in scientific notation: \( 0.00013 = 1.3 \times 10^{-4} \).
4. Apply the logarithm property: \( \ln(1.3 \times 10^{-4}) = \ln(1.3) + \ln(10^{-4}) \).
5. Calculate \( \ln(10^{-4}) \) using the property \( \ln(10^b) = b \cdot \ln(10) \), which simplifies to \( -4 \cdot \ln(10) \).

Key Concepts

Natural Logarithm (ln)

The natural logarithm, denoted as 'ln', is the logarithm to the base 'e', where 'e' is approximately equal to 2.71828. It is used to solve equations involving exponential growth or decay. The natural logarithm is particularly useful in calculus and in solving problems related to continuous compounding.

Properties of Logarithms

Logarithms have several key properties that simplify calculations, such as the product, quotient, and power rules. For instance, ln(a * b) = ln(a) + ln(b) and ln(a/b) = ln(a) - ln(b). Understanding these properties is essential for manipulating logarithmic expressions and solving equations involving logarithms.

Approximation and Rounding

When dealing with logarithmic values, especially in practical applications, it is often necessary to approximate results to a certain number of decimal places. Rounding to four decimal places means adjusting the number to the nearest ten-thousandth, which is important for clarity and precision in reporting results in scientific and mathematical contexts.