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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1:40 minutes

Problem 31

Textbook Question

Use a calculator to find an approximation to four decimal places for each logarithm. ln 144,000

Verified step by step guidance

Step 1: Understand that \( \ln \) represents the natural logarithm, which is the logarithm to the base \( e \), where \( e \approx 2.71828 \).

Step 2: Recognize that you need to find \( \ln(144,000) \), which is the power to which \( e \) must be raised to equal 144,000.

Step 3: Use a scientific calculator to input \( \ln(144,000) \).

Step 4: Ensure your calculator is set to the correct mode for natural logarithms, often labeled as 'ln'.

Step 5: Read the result from the calculator, which should be rounded to four decimal places.

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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 an irrational constant approximately equal to 2.71828. It is commonly used in mathematics, particularly in calculus and exponential growth models. The natural logarithm helps in solving equations involving exponential functions and is essential for understanding continuous growth processes.

Logarithmic Properties

Logarithmic properties are rules that simplify the manipulation of logarithmic expressions. Key properties include the product rule (ln(a*b) = ln(a) + ln(b)), the quotient rule (ln(a/b) = ln(a) - ln(b)), and the power rule (ln(a^b) = b*ln(a)). Understanding these properties is crucial for breaking down complex logarithmic calculations into simpler components.

Calculator Functions

Most scientific calculators have a dedicated function for calculating natural logarithms, typically labeled as 'ln'. To find the natural logarithm of a number, you simply input the number and press the 'ln' button. Familiarity with using a calculator effectively is essential for accurately computing logarithmic values, especially when precision to a specific number of decimal places is required.