Question

Evaluate the logarithm using a calculator. Round your answer to three decimal places. \(\log 6\)

Short answer

The value of \( \log 6 \) rounded to three decimal places is 0.778.

Step-by-step solution

Use a Calculator

Use a calculator with logarithm functionality. Enter the value 6 and then press the 'log' button to find the base-10 logarithm of 6.

Round the Result

The calculator's output will be a decimal number, round this number to three decimal places.

Key concepts

Logarithmic Functions

Logarithmic functions are a fundamental concept in mathematics, often used to simplify complex multiplicative operations. Essentially, these functions are the inverse of exponential functions. If you have an equation in the form of \( y = b^x \), a logarithmic function helps you determine the power \( x \) needed for \( b \) to be raised in order to get \( y \). It is expressed as \( x = \log_b(y) \).

To get a better understanding, think of logarithms as the answer to the question: "To what exponent must the base be raised, to yield a certain number?" Logarithmic functions have several practical applications, especially in fields such as science, engineering, and finance.

Base-10 Logarithm

Among the various logarithmic bases, the base-10 logarithm, commonly denoted as \( \log \) or \( \log_{10} \), is particularly widespread. This type of logarithm uses 10 as its base, which reflects the decimal system we use daily. For instance, \( \log_{10}(100) \) equals 2, because \( 10^2 \) equals 100.

Base-10 logarithms simplify many calculations, such as those involved in scientific notation and decibel measurements. They help by converting multiplicative relationships into additive ones, making calculations easier and more intuitive. Calculators are often programmed with a button dedicated to base-10 logarithms due to their frequent use in problem-solving scenarios.

Using a Calculator

Calculators are indispensable tools for solving mathematical problems, particularly when dealing with logarithms. Most scientific calculators feature a \( \log \) button, which specifically handles base-10 logarithms. Here's how you can use it:

• Insert the number whose logarithm you wish to calculate, such as "6" in this exercise.
• Press the \( \text{log} \) button to get the base-10 logarithm of that number.

A calculator instantly computes the expression \( \log_{10}(6) \), providing a numerical result that corresponds to the power of 10 needed to achieve the entered number. Understanding how to use this function on a calculator is essential for performing more complex computations in various subjects.

Rounding Decimals

Rounding decimals is a practical skill for providing a simplified version of a result without extensive decimal places, often necessary for clarity and comprehensibility. When calculating logarithms, you'll often end up with lengthy decimal numbers that need rounding.

To round a number to three decimal places:

• Identify the fourth decimal place.
• If this digit is 5 or greater, increase the third decimal place by one.
• If it's less than 5, keep the third decimal place as is.

So, if your calculator shows \( 0.7781512 \) as a result for \( \log_{10}(6) \), you round it to \( 0.778 \). This rounding technique is widely used to maintain precision while managing information with minimal complexity.