Question

Use a calculator to evaluate the logarithm. Round your result to three decimal places.\(\log _{10} 345\)

Short answer

The answer, rounded to three decimal places, should be 2.538.

Step-by-step solution

Understand the operation

The problem is asking for a logarithm calculation. This means the aim is to find the exponent that the base, 10, must be raised to in order to get 345. Reiterate that the logarithm base 10 means that if \(10^x = 345\), then \(x = \log _{10} 345\).

Perform the calculation

Using a scientific calculator, input the problem as \(\log 345\). Please note that on most scientific calculators, if no base is specified, it assumes the base is 10.

Round the result

The calculator will give a long decimal number. Since the question requires rounding to three decimal places, round off the answer accordingly. Ensure you remember the rules of rounding: if the digit at the next place value is 5 or more, round up; otherwise, keep the digit the same.

Key concepts

Calculating Logarithms

Understanding how to calculate logarithms is crucial for solving many mathematical problems. In essence, a logarithm answers the question: To what power must we raise a certain base number to produce a given number? In the provided exercise, we're looking for the power to which the base 10 must be raised to yield 345. Formally stated, this is represented as finding the value of x in the equation \(10^x = 345\).

The logarithmic function is the inverse of the exponential function, so the logarithm of a number is the exponent to which the base must be raised. The base-10 logarithm, also known as the common logarithm, is so prevalent in calculations that it is often referred to simply as \(\log\) without explicitly writing the base. When you see \(\log 345\), it is understood that the base is 10 unless otherwise specified.

Understanding the relationship between exponents and logarithms is the key to mastering logarithmic calculations. Remember that \(\log_b a = x\) translates to \(b^x = a\), where b is the base, a is the result we aim for, and x is the exponent or the logarithm we are calculating.

Scientific Calculator Usage

A scientific calculator is a valuable tool for computing complex mathematical problems, including logarithms. When using a scientific calculator to find the value of \(\log _{10} 345\), you should first locate the logarithm function, often labeled as \(\log\) or sometimes as \(\log_{10}\).

Here's a simple guide on how to use the calculator for the given problem:

Entering the Problem

On most scientific calculators, when you press the \(\log\) button and then enter the number 345, the calculator automatically assumes a base of 10. Therefore, you would input it exactly as shown in the problem: \(\log 345\).

Interpreting the Results

After pressing the equal sign or 'Enter', the calculator provides the logarithmic value. This is the exponent you're looking for, and it will usually be displayed as a decimal.

Scientific calculators are designed to handle a broad range of functions, including this one. Familiarity with your particular calculator's functions and layout is essential for efficient problem-solving in subjects such as mathematics, physics, and engineering.

Rounding Decimal Places

Rounding decimal places correctly is a fundamental skill in mathematics and is especially important when the final result needs to be presented with a specific precision. When an exercise asks to round the result to three decimal places, this means you should look at the fourth decimal place to decide whether to round up or stay the same.

Here's a quick guide on how to round:

• Identify the third decimal place in the number; this is your 'target digit'.
• Look to the next digit, the fourth decimal place; this is your 'decision digit'.
• If the decision digit is 5 or more, increase the target digit by one (this is called rounding up).
• If the decision digit is less than 5, leave the target digit unchanged (this is called rounding down).

Let's apply this to our problem: If the calculator shows the result as 2.53794, for instance, we look at the fourth decimal place, which is 9 in this case. Clearly, 9 is greater than 5, so we round up the third decimal place from 3 to 4, resulting in 2.538 when rounded to three decimal places.

Comprehending and applying rounding rules is essential for precise communication in science, technology, engineering, and mathematics fields, where appropriate levels of accuracy are vital.