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

Introduction to Logarithms

College Algebra

6. Exponential & Logarithmic Functions

Introduction to Logarithms

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2:44 minutes

Problem 99

Textbook Question

Given that log￬10 2 ≈ 0.3010 and log￬10 3 ≈ 0.4771, find each logarithm without using a calculator. log￬10 √30

Verified step by step guidance

Recognize that \( \log_{10} \sqrt{30} \) can be rewritten using the property of logarithms: \( \log_{10} (30^{1/2}) = \frac{1}{2} \log_{10} 30 \).

Express 30 as a product of its prime factors: \( 30 = 2 \times 3 \times 5 \).

Use the property of logarithms that states \( \log_{10} (a \times b) = \log_{10} a + \log_{10} b \) to expand \( \log_{10} 30 \) as \( \log_{10} 2 + \log_{10} 3 + \log_{10} 5 \).

Substitute the given values: \( \log_{10} 2 \approx 0.3010 \) and \( \log_{10} 3 \approx 0.4771 \). Note that \( \log_{10} 5 \) can be found using \( \log_{10} 10 = 1 \) and \( \log_{10} 2 + \log_{10} 5 = 1 \), so \( \log_{10} 5 = 1 - \log_{10} 2 \).

Combine all the values and multiply by \( \frac{1}{2} \) to find \( \log_{10} \sqrt{30} \).

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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), while the quotient property states that log_b(m/n) = log_b(m) - log_b(n). Additionally, the power property states that log_b(m^k) = k * log_b(m). These properties allow us to break down complex logarithmic expressions into simpler components.

Change of Base Formula

The change of base formula allows us to express logarithms in terms of logarithms of a different base. Specifically, log_b(a) can be calculated as log_k(a) / log_k(b) for any positive k. This is particularly useful when we have logarithm values for specific bases, such as base 10, and need to compute logarithms for other bases or expressions.

Square Roots and Exponents

The square root of a number can be expressed as an exponent of 1/2. For example, √30 can be rewritten as 30^(1/2). This is important in logarithmic calculations because it allows us to apply the power property of logarithms, enabling us to simplify log_10(√30) to (1/2) * log_10(30), which can then be further broken down using the product property.