Problem 57 - Precalculus

Question

Evaluate the logarithms using the change-of-base formula. Round to four decimal places. $$\log \sqrt{3} 8$$

Step-by-Step Solution

Answer

\( \log_{\sqrt{3}} 8 \approx 3.785 \)

Step 1: Understand the Change-of-Base Formula

The change-of-base formula allows us to rewrite a logarithm in terms of common or natural logarithms. The formula is: \( \log_b a = \frac{\log_c a}{\log_c b} \), where \( c \) is a base of your choice, often 10 or \( e \).

Step 2: Express the Given Logarithm

We want to evaluate \( \log_{\sqrt{3}} 8 \) using the change-of-base formula. Choose base 10 for convenience, so it becomes: \( \frac{\log_{10} 8}{\log_{10} \sqrt{3}} \).

Step 3: Calculate \( \log_{10} 8 \)

Use a calculator to find \( \log_{10} 8 \). This yields approximately 0.9031 when rounded to four decimal places.

Step 4: Calculate \( \log_{10} \sqrt{3} \)

Recall that \( \sqrt{3} = 3^{0.5} \), therefore, \( \log_{10} \sqrt{3} = \log_{10} 3^{0.5} = 0.5 \cdot \log_{10} 3 \). Compute \( \log_{10} 3 \), which is approximately 0.4771. Hence, \( \log_{10} \sqrt{3} \approx 0.2385 \).

Step 5: Divide the Logarithms

Using the change-of-base formula, divide the two results: \( \frac{0.9031}{0.2385} \), which equals approximately 3.785 when rounded to three decimal places.

Key Concepts

Exploring Logarithms

Logarithms are mathematical functions that help us understand how many times we need to multiply a number to get another number. They are the inverse operations of exponentiation. In simple terms, if you know that \( b^x = a \), then \( \log_b a = x \). This tells you "how many times does \( b \) need to be used as a factor to equal \( a \)".
Key points about logarithms:

• The base of a logarithm is the number that is repeatedly multiplied in exponentiation.
• The argument of a logarithm is the result of the base raised to some power.
• Common bases for logarithms are 10 (common logarithm) and \( e \) (natural logarithm).

Logarithms are used extensively in various fields, including science and engineering, to simplify calculations involving large numbers or exponential growth.

Understanding Rounding

Rounding is an essential concept in mathematics that allows us to approximate numbers to make them easier to work with. It's about reducing the digits of a number while keeping its value close to what it was. In the problem above, we need to round to four decimal places, which means the answer should have four digits after the decimal point.How to round a number:

• If the digit after your rounding place is less than 5, round down.
• If it's 5 or more, round up.
• In this problem, numbers are approximated using up to four decimal places, such as 0.9031 for \( \log_{10} 8 \).

Rounding is particularly helpful in ensuring that calculations remain manageable and results communicate intuitive and clear insights, especially when exact numbers are not required.

The Base 10 Logarithm

The base 10 logarithm, often called the common logarithm, is a powerful tool that finds frequent use due to its simplicity and connection with the decimal system. It is denoted as \( \log_{10} \) or just \( \log \) when the base is understood to be 10.
Why use base 10 logarithms?

• They align well with the decimal number system, making them intuitive for calculations on paper.
• Many calculators and software display logarithms with base 10 by default.
• They simplify logarithmic expressions and are often used in scientific notation.

In our example, to evaluate \( \log_{\sqrt{3}} 8 \), we leveraged the base 10 logarithms because they can easily be computed using calculators, showing values like 0.9031 for \( \log_{10} 8 \) and 0.2385 for \( \log_{10} \sqrt{3} \). By dividing these, the result is accurate and precise, which illustrates why base 10 logarithms are a staple in complex calculations.