Problem 47 - A Graphical Approach to College Algebra

Question

Use a calculator to find a decimal approximation for each common or natural logarithm. $$\log 0.783$$

Step-by-Step Solution

Answer

The decimal approximation of \(\log 0.783\) is approximately -0.1063.

Step 1: Understanding Logarithms

The logarithm \(\log 0.783\) indicates the power to which the base 10 must be raised to obtain 0.783. In our context, we want to find this power using a calculator.

Step 2: Input Value in Calculator

Turn on your calculator and locate the logarithm function, usually labeled as "log". Now, input the number 0.783 into the calculator.

Step 3: Calculation

Press the log button followed by the equal sign or enter button. Your calculator computes the logarithm value and shows the results.

Step 4: Approximate the Result

The calculator should display a decimal value. Note that this is an approximation due to the nature of irrational numbers like logarithms. The result should be approximately -0.1063 when rounded to four decimal places.

Key Concepts

Decimal Approximation

When dealing with logarithms like \( \log 0.783 \), the result is often an irrational number. This means the numbers go on infinitely without repeating. Calculators help provide a practical solution by showing a decimal approximation of this value.
This approximation allows us to work with irrational results in a manageable form. Instead of expressing the entire infinite sequence of numbers:

• We round to a specific number of decimal places, like four or five, depending on the needed precision.
• For \( \log 0.783 \), the answer is approximately \(-0.1063\) when rounded to four decimal places.
• Remember, while the approximation is close, it's not exact, but it's sufficient for most purposes.

By understanding decimal approximation, we make complex calculations easier to handle and interpret.

Calculator Usage

Calculators are essential tools for finding logarithms quickly and accurately. Understanding how to use them is key to solving problems like \( \log 0.783 \). Here's a step-by-step guide:

• Turn on your calculator and find the "log" function. This is usually a button labeled with 'log'.
• Input the number you want to find the logarithm for. In this case, it's \( 0.783 \).
• Press the "log" button. This tells the calculator to perform the logarithmic operation.
• Finally, press the equal (=) sign or "enter" to get the result.

With these simple steps, calculators convert complex logarithmic expressions into understandable decimal approximations.
Knowing how to correctly use a calculator for logarithms can streamline your problem-solving process.

Common Logarithm

Common logarithms, denoted as \( \log \), have base 10. They help solve everyday problems related to growth rates, exponential functions, and scientific calculations.
The expression \( \log 0.783 \) looks for the power to which 10 must be raised to yield 0.783. Here's why common logarithms matter:

• They simplify problems involving multipliers and growth rates.
• Used frequently in scientific fields for measurements and calculations.
• Easy to handle with a simple calculator function labeled "log".

Common logarithms make it easy to switch between multiplicative and additive understanding of changes.
This is essential in areas like finance, engineering, and data analysis, where interpreting exponential relationships is crucial.