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Ch. 4 - Inverse, Exponential, and Logarithmic Functions
Lial - College Algebra 13th Edition
Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook
All textbooksLial 13th EditionCh. 4 - Inverse, Exponential, and Logarithmic FunctionsProblem 93
Chapter 5, Problem 93

Given that log10 2 ≈ 0.3010 and log10 3 ≈ 0.4771, find each logarithm without using a calculator. log10 6

Verified step by step guidance
1
Recognize that the logarithm of 6 can be expressed in terms of the logarithms of its prime factors. Since 6 = 2 × 3, write the expression as \(\log_{10} 6 = \log_{10} (2 \times 3)\).
Use the logarithm product rule, which states that \(\log_b (xy) = \log_b x + \log_b y\), to rewrite the expression as \(\log_{10} 6 = \log_{10} 2 + \log_{10} 3\).
Substitute the given approximate values into the expression: \(\log_{10} 2 \approx 0.3010\) and \(\log_{10} 3 \approx 0.4771\).
Add the two logarithm values together: \(0.3010 + 0.4771\).
The result of this addition will give you the approximate value of \(\log_{10} 6\) without using a calculator.

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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 specific properties that simplify calculations, such as the product rule: log_b(mn) = log_b(m) + log_b(n). This allows breaking down complex logarithms into sums of simpler ones, which is essential for solving log_10 6 using known values of log_10 2 and log_10 3.
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Common Logarithms (Base 10)

Common logarithms use base 10 and are often denoted as log or log_10. Understanding that log_10 10 = 1 and how to interpret logarithms in base 10 helps in applying given approximate values like log_10 2 and log_10 3 to find other logarithms.
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Graphs of Common Functions

Approximation and Estimation

Using given approximate values of logarithms allows estimation without a calculator. This skill involves substituting known logarithm values into formulas and performing arithmetic operations to find the desired logarithm accurately.
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