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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1:43 minutes

Problem 93

Textbook Question

Given that log￬10 2 ≈ 0.3010 and log￬10 3 ≈ 0.4771, find each logarithm without using a calculator. log￬10 6

Verified step by step guidance

Recognize that 6 can be expressed as a product of 2 and 3, i.e., 6 = 2 \times 3.

Use the logarithmic property for products: \( \log_{10}(a \times b) = \log_{10} a + \log_{10} b \).

Apply the property to \( \log_{10} 6 \): \( \log_{10} 6 = \log_{10} (2 \times 3) = \log_{10} 2 + \log_{10} 3 \).

Substitute the given values: \( \log_{10} 2 \approx 0.3010 \) and \( \log_{10} 3 \approx 0.4771 \).

Add the two values: \( \log_{10} 6 \approx 0.3010 + 0.4771 \).

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Key Concepts

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

Properties of Logarithms

Logarithms have several key properties that simplify calculations. The product property states that log_b(mn) = log_b(m) + log_b(n), allowing us to break down logarithms of products into sums. Similarly, the quotient property states that log_b(m/n) = log_b(m) - log_b(n), and the power property states that log_b(m^k) = k * log_b(m). These properties are essential for manipulating logarithmic expressions.

Change of Base Formula

The change of base formula allows us to compute logarithms in different bases. It states that log_b(a) = log_k(a) / log_k(b) for any positive k. This is particularly useful when we have logarithm values in one base and need to find values in another base, as seen in the question where we use known logarithm values to find log_10(6).

Logarithm of a Product

To find log_10(6), we can express 6 as a product of its prime factors: 6 = 2 * 3. Using the properties of logarithms, we can then calculate log_10(6) as log_10(2) + log_10(3). By substituting the given approximate values for log_10(2) and log_10(3), we can easily find the logarithm of 6 without a calculator.