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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1:30 minutes

Problem 59

Textbook Question

Find each value. If applicable, give an approximation to four decimal places. See Example 5. ln 27 + ln 943

Verified step by step guidance

Recognize that the problem involves the natural logarithm function, denoted as \( \ln \).

Use the property of logarithms that states \( \ln a + \ln b = \ln (a \times b) \).

Apply this property to combine the logarithms: \( \ln 27 + \ln 943 = \ln (27 \times 943) \).

Calculate the product inside the logarithm: \( 27 \times 943 \).

Once the product is calculated, find the natural logarithm of the result to get the final value.

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Key Concepts

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

Natural Logarithm

The natural logarithm, denoted as 'ln', is the logarithm to the base 'e', where 'e' is approximately equal to 2.71828. It is a fundamental concept in algebra and calculus, often used to solve equations involving exponential growth or decay. Understanding how to manipulate natural logarithms is essential for simplifying expressions and solving equations.

Properties of Logarithms

Logarithms have several key properties that simplify calculations. One important property is that the sum of two logarithms with the same base can be expressed as the logarithm of the product of their arguments: ln(a) + ln(b) = ln(ab). This property is crucial for solving problems involving multiple logarithmic terms, as it allows for the combination of terms into a single logarithm.

Approximation and Rounding

In many mathematical contexts, especially when dealing with logarithmic values, approximation is necessary. Rounding to a specified number of decimal places, such as four in this case, helps in presenting results clearly and concisely. Understanding how to round numbers correctly is important for accuracy in calculations and for meeting specific formatting requirements in mathematical problems.