Problem 2 - Algebra A Combined Function

Question

Use a calculator to approximate each logarithm to four decimal places. $$ \log 6 $$

Step-by-Step Solution

Answer

\( \log 6 \approx 0.7782 \)

Step 1: Understand Logarithms

A logarithm is the inverse operation of exponentiation. It helps us determine the exponent needed for a base number to reach a certain value. When you see \( \log 6 \), it means 'to what power must 10 be raised, to obtain 6?' The base of the common logarithm is always 10.

Step 2: Use Your Calculator

To find \( \log 6 \) using a calculator, locate the 'log' button, which typically represents the base-10 logarithm. Enter '6' on your calculator and then press the 'log' button.

Step 3: Read and Round the Result

After entering '6' and pressing 'log', your calculator will display the result. The answer will likely be a decimal number. Round this result to four decimal places to match the problem's requirement.

Step 4: Write the Final Answer

Once you have rounded the decimal result from Step 3 to four decimal places, write this as your final answer for \( \log 6 \). The result should be precise to four digits after the decimal.

Key Concepts

Exponentiation

Exponentiation is a fundamental mathematical concept where a number, known as the base, is raised to the power of an exponent. This operation is symbolized as \( b^n \), where \( b \) is the base, and \( n \) is the exponent or the power. In simpler terms, exponentiation involves multiplying the base number by itself as many times as indicated by the exponent.
For example:

• \( 2^3 = 2 \times 2 \times 2 = 8 \)
• \( 5^4 = 5 \times 5 \times 5 \times 5 = 625 \)

Exponentiation is an essential building block for understanding logarithms, as a logarithm is the inverse of this process. If you have a specific number that results from exponentiating a base, using logarithms can help you find the exponent used.

Base 10 Logarithm

The base 10 logarithm, often called the common logarithm, is one of the most frequently encountered types of logarithms. It is represented simply as \( \log \). When you read \( \log 6 \), it specifically asks, "To what power must 10 be raised to get 6?" In mathematical terms, it's asking for the exponent in the equation \( 10^x = 6 \).
Base 10 logarithms are widely used because the decimal system is based on powers of 10. They simplify many calculations, especially when dealing with very large or very small numbers. For any number \( x \), \( \log x \) explores the relationship between exponential and logarithmic forms:

• \( 10^{\log x} = x \)
• If \( x = 1000 \), then \( \log 1000 = 3 \)

The base 10 logarithm provides insights into exponential growth patterns and is vital in fields like science, engineering, and finance.

Calculator

For those solving logarithmic problems without paper and pen, a calculator is an invaluable tool. Most scientific calculators are equipped with a 'log' button, which is specifically designed for base 10 logarithms. Here's a simple guide to using a calculator to find the base 10 logarithm, like \( \log 6 \):

• Locate the 'log' button on your calculator.
• Input the number for which you need to compute the logarithm (in our exercise, it's 6).
• Press the 'log' button, and the calculator will immediately display the logarithmic value.
• View the decimal number displayed (usually with more than four digits after the decimal point).
• Round the result to four decimal places to obtain the precise answer required.

Utilizing a calculator simplifies complex calculations and ensures accuracy when working with logarithms, letting you focus more on understanding the problem rather than just the mechanics of calculation.