Problem 89 - Intermediate Algebra

Question

Use a calculator to find each value. Give answers to four decimal places. See Using Your Calculator: Evaluating Logarithms. $$ \log 0.00467 $$

Step-by-Step Solution

Answer

The logarithm of 0.00467 is approximately -2.3304.

Step 1: Understand the Problem

The task is to evaluate the logarithm of the number 0.00467. This is asking for the power to which 10 must be raised to produce 0.00467.

Step 2: Use a Calculator

Turn on your calculator and ensure it's in log mode (base 10 logarithm).

Step 3: Enter the Value into the Calculator

Enter the number 0.00467 into your calculator to find the logarithm value. Make sure the number is input correctly.

Step 4: Calculate and Round

Press the log key (usually 'log' or 'LOG') on your calculator, followed by '=' or 'Enter' to get the result. The value should be approximately -2.3304 when rounded to four decimal places.

Key Concepts

Understanding Logarithm Calculation

Logarithm calculation is an essential skill in mathematics, particularly when dealing with exponential relationships. A logarithm answers the question: "What power is a certain base raised to in order to obtain a specific number?" In this problem, you are calculating \( \log 0.00467 \), which asks you to find the power you need to raise 10 to, in order to get 0.00467. This requires you to transform the number 0.00467 into an exponent of 10.

• The base of this logarithm is 10, known as the common logarithm.
• The task involves finding the \'exponent\' since logarithms are the inverses of exponentials.
• When you calculate, you find that \( 10^{-2.3304} \) approximately equals 0.00467, explaining the result of \( -2.3304 \).

Using calculators simplifies this process by providing direct computation of the logarithmic value.

Using a Calculator Efficiently for Logarithms

Using a calculator is crucial for solving logarithms, especially when high precision is required. Most scientific calculators have a dedicated 'log' button for base 10 logarithms, which simplifies the process significantly. Here’s a quick guide:

• Turn on your calculator and set to its standard calculation mode if needed.
• Input the number you want the logarithm of, in this case, 0.00467.
• Press the 'log' button. This automatically uses base 10 for calculation.
• Hit the '=' or 'Enter' button to get the result.

The somewhat tricky part might be ensuring you enter the number correctly, so always double-check your inputs. Rounding your answer to the desired decimal places, four in this case, will help you meet problem requirements.

Demystifying Base 10 Logarithms

Base 10 logarithms, also known as common logarithms, are widely used in many fields, from science to engineering. These logarithms use 10 as their base, implying they are answering the question, "How many times do we multiply 10 to get our number?" This is very helpful with our decimal number system, making it particularly easy to interpret and apply.

• Base 10 allows a straightforward correlation with powers of 10, making it intuitive.
• On calculators, the 'log' button corresponds to a base 10 logarithm by default.
• These logarithms simplify calculations involving very large or very small numbers.

Due to their simplicity and the ability to convert multiplication into addition, base 10 logarithms are a preferred tool for handling complex calculations in everyday technological applications.