Problem 15 - Algebra 2

Question

Use a calculator to evaluate each expression to four decimal places. $$ \log 12 $$

Step-by-Step Solution

Answer

1.0792

Step 1: Recall the Logarithm Base

The logarithm function without an explicitly stated base assumes the base is 10. Thus, we are tasked with calculating \( \log_{10} 12 \).

Step 2: Use the Calculator

Input 12 into your calculator and then choose the logarithm function (commonly labeled \( \log \) on calculators, meaning base 10).

Step 3: Read the Calculator Output

After pressing the \( \log \) button, the calculator will display the value of \( \log_{10} 12 \). Ensure the calculator is set to display four decimal places.

Step 4: Write Down the Result

Record the result from the calculator as 1.0792, which is \( \log_{10} 12 \) rounded to four decimal places.

Key Concepts

logarithmic functions

Logarithmic functions are essential in mathematics, used to solve equations involving exponential growth or decay. A logarithm answers the question: "To what power should the base be raised to produce a given number?" For example, in the expression \( \\log_b (x) = y \), \( b^y = x \). Here, \( b \) is the base, \( x \) is the number we are finding the logarithm of, and \( y \) is the exponent that the base is raised to produce \( x \). Logarithmic functions are the inverse of exponential functions and are widely used in fields like engineering, science, and finance.
Understanding their properties can help in analyzing and interpreting data, especially when comparing rates of growth and decay across different scales.

base 10 logarithms

Base 10 logarithms, often known as common logarithms, use 10 as the base and are represented as \( \log (x) \) without a specific base indicator. This is because 10 is the most common base used in everyday calculation, especially in scientific and engineering calculations, where it helps in simplifying multiplication and division into addition and subtraction. For instance, \( \log_{10} 100 = 2 \) because \( 10^2 = 100 \).
In practical applications, base 10 helps in understanding phenomena that involve orders of magnitude, like population growth and sound intensity. When using a calculator, the button labeled \( \log \) typically represents the base 10 logarithm, making it straightforward to perform such calculations without manually raising 10 to various powers.

calculator usage

Using a calculator to find logarithmic values is a convenient and efficient method. To calculate \( \log_{10} 12 \), start by turning on your calculator. Then, input the number 12 by pressing the appropriate keys. Locate the \( \log \) button, which stands for base 10 logarithm. Press this button and the calculator will compute the logarithmic value. Make sure your calculator is set properly to ensure accuracy in your calculation.
Calculators simplify the process by doing the exponentiation and inversion required in logarithmic calculations, providing quick results that would otherwise require significant time and manual computation.

decimal place rounding

Decimal place rounding involves reducing the number of decimal places in a number while maintaining its representation. It's crucial for ensuring consistency and readability in presentations of numerical data.

• Identify the desired number of decimal places—in this case, four.
• Check the digit immediately after your desired cutoff point. If it's five or above, round up the last kept digit. Otherwise, leave it.

For example, when evaluating \( \log_{10} 12 \) with a result like 1.079181, we round this to four decimal places to get 1.0792. Rounding appropriately helps in maintaining fixed precision in calculations, essential in fields where slight changes can lead to significant differences.