Problem 36 - Precalculus : Building Concepts and Connections

Question

Evaluate the expression to four decimal places using a calculator. $$-3 \log 6$$

Step-by-Step Solution

Answer

The result of the expression \(-3 \log 6\) to four decimal places is -2.7781.

Step 1: Understand the logarithm operation

The expression \(-3 \log 6\) involves the logarithm operation. In general, the logarithm base 10 of a number is the exponent to which 10 must be raised to get the number. The expression \(-3 \log 6\) means that the logarithm of 6 is found first and then multiplied by -3.

Step 2: Use a scientific calculator

With the understanding of logarithm operation, a scientific calculator can be used to get the accurate result. For this operation, first type '6', then press the 'log' button, and finally multiply the result by '-3'.

Step 3: Round off the result to four decimal places

The result from a calculator is often between 8 to 10 decimal places. Rounding off the result to four decimal places as per the exercise gives the final answer.

Key Concepts

Scientific Calculator Usage

When tackling mathematical problems that involve logarithms, a scientific calculator becomes your best ally. These calculators are equipped with functions that can compute the logarithm of a number with precision. Here's how you can use a scientific calculator for such computations:

• Begin by turning on your calculator and ensuring it's set to calculate base 10 logarithms. This is usually the default setting.
• Input the number whose logarithm you wish to calculate; in this case, type in '6'.
• Press the 'log' button, typically labeled with "LOG" or "log" on the calculator. This button applies the base 10 logarithm to the entered number.
• The display will show the logarithm of the number you entered.

After taking the logarithm, multiply the result by any necessary coefficient, in this task, by -3. This feature yields efficient and accurate results, thus saving time and reducing errors in manual computations.

Logarithmic Operations

Logarithmic operations are a fundamental aspect of mathematics, designed to solve equations involving exponential growth or decay. In this exercise, the expression \(-3 \log 6\) combines both multiplication and logarithms. To understand this:

• Logarithms answer the question: "To what power must the base (in this case, 10) be raised to produce a certain number (here, 6)?"
• The expression \( \log 6 \) computes the base 10 logarithm of 6.
• Once this value is determined, multiply it by -3 as stated in the expression.

This operation effectively scales the logarithmic value by -3, altering its magnitude significantly. Logarithmic calculations are essential for a wide range of applications, including scientific data analysis, financial forecasts, and more.

Rounding Decimal Places

In mathematical calculations, especially when dealing with non-integer results, rounding is a crucial skill. Rounding decimal places makes outcomes more manageable and often aligns with given precision requirements. Here's how to round to four decimal places:

• Start with the full calculator result. If it's displayed as, say, 2.558742, you'll need to evaluate it for rounding.
• Focus on the fifth decimal digit, as it determines whether the fourth decimal digit increases by one.
• If the fifth digit is 5 or more, round up the fourth digit. If it's less than 5, keep the fourth digit as is.
• For example, in 2.5587, you observe the next digit is 4 (less than 5), so the rounded number is 2.5587.

Rounding ensures clarity and precision; an important aspect in conveying concise and error-free data across various disciplines.