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

Ch. 4 - Inverse, Exponential, and Logarithmic Functions

Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook

All textbooks

Lial 13th Edition

Ch. 4 - Inverse, Exponential, and Logarithmic Functions

Problem 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

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.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

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.

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.

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.

Recommended video: