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Chapter 4: Problem 36

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

Short Answer

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

Step by step solution

01

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.
02

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'.
03

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

Solve using any method, and eliminate extraneous solutions. $$\ln |2 x-3|=1$$

The value of a 2006 S-type Jaguar is given by the function $$v(t)=43,173(0.8)^{t}$$ where \(t\) is the number of years since its purchase and \(v(t)\) is its value in dollars. (Source: Kelley Blue Book) (a) What was the Jaguar's initial purchase price? (b) What percentage of its value does the Jaguar S-type lose each year? (c) How many years will it take for the Jaguar S-type to reach a value of \(\$ 22,227 ?\)

The function \(f(x)=x^{6}\) is not one-to-one. How can the domain of \(f\) be restricted to produce a one-to-one function?

The following data gives the percentage of women who smoked during pregnancy for selected years from 1994 to \(2002 .\) (Sournce: National Center for Health Statistics) $$\begin{array}{|c|c|} \hline\text { Year } & \text { Percent Smoking } \\\\\text { Yuring Pregnancy } \\ \hline1994 & 14.6 \\\1996 & 13.6 \\\1998 & 12.9 \\\2000 & 12.2 \\\2001 & 12.0 \\\2002 & 11.4\\\\\hline\end{array}$$ (a) From examining the table, what is the general relationship between the year and the percentage of women smoking during pregnancy? (b) Let \(t\) be the number of years after \(1993 .\) Here, \(t\) starts at 1 because in 0 is undefined. Make a scatter plot of the data and find the natural logarithmic function of the form \(p(t)=a \ln t+b\) that best fits the data. Why must a be negative? (c) Project the percentage of women who will smoke during pregnancy in the year 2007.

Give an example of an odd function that is not one-to-one.
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