Question

Use a calculator to evaluate the logarithm. Round your result to three decimal places.\(\log _{10} \sqrt{8}\)

Short answer

The value of \( \log_{10} \sqrt{8} \) rounded to three decimal places is 0.452.

Step-by-step solution

Find the exponent of 8 under the square root

Since a square root is represented by an exponent of 0.5, you have: \( \sqrt{8} = 8^{0.5} \). So, the equation from the exercise may be rewritten as \( \log_{10} 8^{0.5} \).

Use the property of logarithms to simplify

Using the property of logarithms that \( \log_b a^k = k*log_b a \), the equation becomes \( 0.5*\log_{10} 8 \).

Calculate log using a calculator

Now use a calculator which is able to calculate base-10 logarithms. Doing this yields \( 0.5*\log_{10} 8 \approx 0.5*0.903=0.4515 \).

Round to three decimal places

Finally, round the result to three decimal places: 0.452.

Key concepts

Properties of Logarithms

The beauty of logarithms lies in their properties, which make complex calculations more manageable. Understanding these properties can greatly simplify the process of evaluating expressions involving logarithms. One fundamental property is the power rule, which states that the logarithm of a number raised to an exponent is the exponent times the logarithm of the number, represented as \( \log_b a^k = k \cdot \log_b a \). This is particularly useful when dealing with square roots, as they can be expressed as exponents, i.e., \( \sqrt{x} = x^{0.5} \).

Other properties include the product rule \( \log_b(mn) = \log_b(m) + \log_b(n) \), and the quotient rule \( \log_b(\frac{m}{n}) = \log_b(m) - \log_b(n) \), which help break down multiplication and division within a logarithm into simpler addition or subtraction operations. The change of base formula is also invaluable when you don't have a calculator set for a specific base, as it allows converting between bases: \( \log_b a = \frac{\log_c a}{\log_c b} \), where \(c\) is the new base.

Calculating Square Roots

Calculating the square root of a number is a fundamental concept in mathematics. The square root of any non-negative number \( x \) can be written as \( x^{0.5} \). While simple square roots of perfect squares can often be determined mentally, square roots of non-perfect squares typically require a calculator.

When working with logarithms, this understanding of square roots as exponents allows us to apply log properties effectively. If the problem involves a logarithm of a square root, as in the exercise \( \log_{10} \sqrt{8} \), rephrasing it using exponents becomes instrumental. This rephrasing aligns with the principle that the more complex an equation looks, the more we should try to break it into smaller, easier to manage pieces by utilizing mathematical properties and rules.

Using Calculators for Logarithms

Calculators are indispensable when it comes to dealing with logarithms, especially when the numbers involved aren't based on common roots or powers. In order to find the logarithm of a number, calculators typically have a button labeled either \( \log \), indicating the common (base 10) logarithm, or \( \ln \), indicating the natural (base \( e \) ) logarithm.

To solve the given exercise, you need to use the \( \log \) function on your calculator. If you're working with a scientific calculator, simply input the number and press the \( \log \) key to get the result. If you are using a more advanced calculator, such as a graphing calculator, you might have the capability to directly input the entire expression \( \log_{10} 8^{0.5} \) and get the answer. Remember to always verify if your calculator is set to the correct mode – degree or radian when dealing with trigonometric functions, as it can affect your logarithm calculations as well. After calculations, rounding off to the desired number of decimal places is critical for precision.