Question

In Exercises, use a calculator to evaluate the logarithm. Round to three decimal places. $$ \log _{3} \frac{1}{2} $$

Short answer

The answer is approximately \(-0.631\).

Step-by-step solution

Identifying the Base and Argument of the Logarithm

The base of the logarithm is the number \(3\). This is the base that we will input into our calculator. The argument of the logarithm is \(1/2\). This is what we're trying to find the logarithm of.

Calculating the Logarithm using a Calculator

The next step is to enter these figures into the calculator. Most calculators will have a button that says 'log' or something similar. This will be used to compute the logarithm. Input \(1/2\) and then the log button on the calculator. Make sure the calculator is set to compute the logarithm base \(3\). Then press equals or calculate to get the answer.

Rounding to Three Decimal Places

The final step is to round the answer from the calculator to three decimal places. This is done because the exercise specifies to round the answer to this decimal place.

Key concepts

Understanding Logarithmic Functions

Logarithmic functions are the inverse functions of exponentials, and they solve the often-asked question, 'To what power must the base be raised to produce a certain number?' The base of a logarithm, as seen in the exercise \( \log _{3} \frac{1}{2} \), is the constant that the variable is being compared to—in this case, \(3\). Understanding this concept is crucial because it lays the groundwork for evaluating logarithms. The argument here, \( \frac{1}{2} \), represents the value we want to find the power for when raised with our base.
Logarithmic functions can be used in various fields like mathematics, physics, and computer science to solve problems involving exponential growth or decay, such as the spread of diseases, radioactive decay, or population growth.
To evaluate a logarithm without a calculator, you would typically use properties of logarithms to simplify the expression into a form that can be calculated manually. However, for non-integer arguments or bases other than \(10\), \(e\), and \(2\), using a calculator is more practical.

Calculator Use in Logarithms

Utilizing a calculator to solve logarithms, particularly when dealing with non-standard bases like \(3\), simplifies the process considerably.
First, ensure your calculator can compute logarithms with bases other than \(10\) and \(e\). If it doesn't, you'll use the change of base formula: \( \log_b a = \frac{\log_c a}{\log_c b} \), often selecting \(10\) or \(e\) for \(c\). Enter the argument followed by the \(\log\) function and, if required, switch to the desired base.
For the exercise in question, if the calculator has an option for different bases, you will enter \(1/2\), choose 'log', input the base \(3\), and then calculate the result. Calculators are designed to handle this operation with precision, which is why they are indispensable for students studying logarithms.

Rounding Decimals in Logarithms

Rounding is a mathematical technique used to reduce the number of decimal places to a more manageable figure without overly compromising accuracy. In the context of logarithms, numerical results can be very lengthy; thus, rounding them to a specified number of decimal places is often required.
The approach for rounding to three decimal places, as specified in the exercise, involves looking at the fourth decimal place. If this digit is \(5\) or higher, you increase the third decimal place by one. If it's lower than \(5\), the third decimal place remains unchanged.
This process ensures that the student's answer is concise and consistent with mathematical conventions, enhancing the readability and making it easier to compare and use in subsequent calculations. Whether using logarithms for practical applications or in an academic setting, accurate rounding ensures clarity and precision.
Always take note of whether the requirement is to round up or down, as this can affect the result slightly but significantly, especially in scientific and financial contexts.