Table of contents

0. Review of Algebra4h 18m
1. Equations & Inequalities3h 18m
2. Graphs of Equations43m
3. Functions2h 17m
4. Polynomial Functions1h 44m
5. Rational Functions1h 23m
6. Exponential & Logarithmic Functions2h 28m
7. Systems of Equations & Matrices4h 6m
8. Conic Sections2h 23m
9. Sequences, Series, & Induction1h 22m
10. Combinatorics & Probability1h 45m

6. Exponential & Logarithmic Functions

Properties of Logarithms

College Algebra

6. Exponential & Logarithmic Functions

Properties of Logarithms

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1:19 minutes

Problem 53

Textbook Question

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

Verified step by step guidance

Recognize that the problem requires finding the natural logarithm of a number, specifically \( \ln(0.00013) \).

Recall that the natural logarithm function, \( \ln(x) \), is the inverse of the exponential function \( e^x \).

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: \( 0.00013 = 1.3 \times 10^{-4} \).

Apply the logarithm property: \( \ln(1.3 \times 10^{-4}) = \ln(1.3) + \ln(10^{-4}) \).

Calculate \( \ln(10^{-4}) \) using the property \( \ln(10^b) = b \cdot \ln(10) \), which simplifies to \( -4 \cdot \ln(10) \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.