Question

Use a calculator to evaluate the logarithm. Round your result to three decimal places.\(\log _{10} \sqrt{3}\)

Short answer

The result of the logarithmic function is approximately \(0.248\).

Step-by-step solution

Rewrite the square root as an exponent

Rewrite \(\sqrt{3}\) as \(3^{0.5}\). So the expression they need to evaluate becomes \(\log _{10} 3^{0.5}\)

Use the property of logarithms

When you apply the property of logarithms \(a \log b = \log b^a\), the result is \(0.5 \log _{10} 3\).

Enter into the calculator

Input the expression \(0.5 \log _{10} 3\) into your calculator and round your result to three decimal places. The answer will be the decimal you obtain.

Key concepts

Logarithms

Understanding logarithms is essential for solving many mathematical problems involving exponential relationships. A logarithm answers the question: To what power must a given base number be raised to obtain a specific number? The expression \( \log_b x \) corresponds to the power or exponent \( y \) in the equation \( b^y = x \). In the exercise \( \log_{10} \sqrt{3} \) we're looking for the power to which 10 must be raised to get \( \sqrt{3} \).

Logarithms come in handy when dealing with large numbers or complex calculations since they transform multiplicative processes into additive ones. This transformation is possible due to the unique properties of logarithms, which will be discussed in a following section. They simplify the calculations and make it easier to understand the underlying relationships. Regular practice with a logarithm calculator will reinforce your comprehension and application of this concept.

Exponential Expressions

Exponential expressions are a way to represent repeated multiplication. The expression \( a^b \) signifies that the base \( a \) is multiplied by itself \( b \) times. In the context of logarithms, understanding exponential expressions is crucial because logarithms are, in essence, the inverse of exponentials.

In the given problem, the square root of 3, \( \sqrt{3} \) can be expressed as \( 3^{0.5} \) because square roots correspond to the one-half power. When you use a logarithm calculator, rewriting square roots or other radical expressions as exponents simplifies the computation. It's crucial to grasp this idea because it consistently appears in problems involving growth and decay, compounding interest, and many areas of science and engineering.

Properties of Logarithms

The properties of logarithms greatly facilitate the process of calculation and allow us to manipulate logarithmic expressions. One of these properties is the power rule, which states that \( \log_b (a^c) = c \cdot \log_b a \). It allows us to move the exponent of the argument to the front of the logarithm, effectively converting a multiplicative relationship into an additive one.

This very property is applied in Step 2 of the textbook solution, where the expression \( \log_{10} 3^{0.5} \) becomes \( 0.5 \cdot \log_{10} 3 \). Utilizing these properties is fundamental for simplifying logarithmic expressions before inputting them into calculators. Recognizing and applying the properties correctly can transform a seemingly complex calculation into a more manageable form, making it easier to input and reducing potential errors during calculation.