Problem 1 - Algebra A Combined Function

Question

Use a calculator to approximate each logarithm to four decimal places. $$ \log 8 $$

Step-by-Step Solution

Answer

\(\log 8 \approx 0.9031\)

Step 1: Understanding Logarithms

A logarithm is the power to which a number must be raised in order to get some other number. So, \(\log 8\) asks 'to what power must we raise the base 10 to get 8?'.

Step 2: Setting Up the Calculation

To find \(\log 8\) using a calculator, ensure that it is set to use base 10, which is the default setting for common logarithms, or \(\log\) calculations.

Step 3: Inputting into the Calculator

Turn on the calculator and press the \(\log\) button, then enter the number 8. It should look like \(\log(8)\) on the display.

Step 4: Calculating the Logarithm

Press the 'equals' or 'enter' button, depending on your calculator model, to perform the calculation of \(\log 8\).

Step 5: Rounding the Result

The calculator will display a result, which should be approximately 0.903. Round this number to four decimal places to get \(\log 8 \approx 0.9031\).

Key Concepts

Using a Calculator

When working with logarithms, using a calculator can greatly simplify the process, especially for common logarithms like \( \log 8 \). Most calculators have a dedicated logarithm function, typically labeled as "log". To use this function correctly:

• Switch on your calculator.
• Locate and press the "log" button. This button is set to calculate base 10 logarithms, which are standard for common log calculations.
• Enter the number for which you need to find the logarithm, in this case, 8.
• Press "equals" or "enter" to compute the result.

Calculators are designed to handle these calculations accurately and quickly. Always double-check which base your calculator is set in when tackling logarithmic functions, especially in non-standard calculations.

Common Logarithms

Common logarithms are a type of logarithm where the base is 10. These are written as \( \log_{10} \) but are commonly abbreviated as \( \log \) without specifying the base because base 10 is the default. Common logarithms answer the question: 'To what power must 10 be raised to yield a given number?'
They are widely used in various science and engineering fields due to their simplicity with the base 10 number system, which aligns with our decimal. For example, \( \log 8 \) is seeking the power to which 10 must be raised to result in 8.
Understanding this concept is key, as it helps demystify why calculators are set up to use base 10 by default. It's important not only for executing the calculation but also for interpreting the result.

Rounding Decimal Values

When you're working with the logarithmic values obtained from a calculator, rounding is often necessary to achieve the desired precision. In mathematics, particularly when dealing with problems that specify a number of decimal places, rounding ensures consistency and accuracy in your results.

• If your result displays a longer series of digits, identify the number of decimal places required—in this exercise, four decimal places are specified.
• Look at the digit immediately following your last desired decimal place. If it is 5 or higher, round the last place up by one. If it is 4 or lower, leave it as is.
• So, \( \log 8 \approx 0.9031 \) is reached by rounding 0.903089987 to four decimal places.

Mastering rounding techniques ensures that your answers are both precise and formatted correctly for scores and exact reporting.