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

Verified step by step guidance

1. Recognize that the problem asks for the natural logarithm of 0.00013, which is written as \(\ln 0.00013\). The natural logarithm function, \(\ln x\), is the logarithm to the base \(e\), where \(e \approx 2.71828\).
2. Rewrite the number 0.00013 in scientific notation to make the logarithm easier to handle: \(0.00013 = 1.3 \times 10^{-4}\).
3. Use the logarithm property that \(\ln(ab) = \ln a + \ln b\) to separate the expression: \(\ln(1.3 \times 10^{-4}) = \ln 1.3 + \ln 10^{-4}\).
4. Recall that \(\ln 10^{-4} = -4 \ln 10\). Since \(\ln 10\) is approximately 2.3026, you can express this part as \(-4 \times 2.3026\) (do not calculate the final value yet).
5. Find or approximate \(\ln 1.3\) using a calculator or logarithm table, then add it to the value from the previous step to get the final natural logarithm value. If required, round your answer to four decimal places.

Key Concepts

Natural Logarithm (ln)

The natural logarithm, denoted as ln, is the logarithm to the base e, where e ≈ 2.71828. It answers the question: to what power must e be raised to get a given number? For example, ln(x) = y means e^y = x. It is defined only for positive real numbers.

Evaluating Logarithms of Small Numbers

When evaluating ln of a small positive number less than 1, the result is negative because e raised to a negative power yields a fraction between 0 and 1. Understanding this helps interpret the sign and magnitude of the logarithm for values like 0.00013.

Decimal Approximation and Rounding

After calculating the exact value of ln(0.00013), it is often necessary to approximate the result to a certain number of decimal places. Rounding to four decimal places means keeping four digits after the decimal point and adjusting the last digit based on the next digit's value.