Question

Use a calculator to evaluate the logarithm. Round your result to three decimal places.\(\log _{10} \frac{4}{5}\)

Short answer

The value of \( \log _{10} \frac{4}{5} \) rounded to three decimal places is -0.097.

Step-by-step solution

Input Calculation

Enter the fraction 4/5 into the calculator, ensuring to place brackets around the fraction to keep it together. You have \( \frac{4}{5} \).

Logarithm Base

Next, enter 'log' into the calculator (most calculators have base 10 as the default logarithm base, but double check your specific calculator).

Calculate Logarithm

Now, you should be able to calculate the base 10 logarithm of the fraction 4/5 by pressing equals. The result you get will have many decimal places.

Round the Result

Finally, round the result you found to three decimal places, as per the instructions in the problem. Use the standard rounding rules, remembering that if the fourth decimal place is 5 or more, you'll round up.

Key concepts

Logarithm Base 10

Understanding logarithm base 10 is crucial when it comes to solving many mathematical problems, especially in fields such as science and engineering. Logarithms are the inverses of exponentiation. A base 10 logarithm, often written as \(\log_{10}\), asks the question: to what power must we raise 10 to obtain a certain number?

For example, if we have \(\log_{10}(100)\), we are looking for the power that 10 must be raised to, to get 100. The answer is 2, because \(10^2 = 100\). This is the essence of a logarithm in base 10. It's applied in the context of the problem at hand, where we need to find the power that 10 must be raised to, to get \(\frac{4}{5}\).

Logarithm base 10 is incredibly common, as it aligns with our decimal number system. It's also referred to as the common logarithm and has use cases spanning from calculating pH in chemistry to measuring the Richter scale in earthquakes.

Rounding Decimal Places

Rounding decimal places is an essential skill across all areas of mathematics and science, ensuring that we can succinctly express numbers without losing important information. When we round to a certain number of decimal places, we look at the digit immediately after our desired precision.

For instance, if we are rounding to three decimal places and our number is 0.12345, we look at the fifth digit (which is a 5). Since it's 5 or higher, we increase the third digit by one, resulting in 0.123. If the fifth digit were lower than 5, we would leave the third digit as is. This method helps maintain accuracy in measurements while keeping numbers manageable.

In the context of the original problem, this means taking the calculator's result for \(\log_{10}\frac{4}{5}\) and applying the rounding rule to three decimal places. It's also crucial to understand this concept as you encounter different scenarios which require varying levels of precision.

Using a Calculator for Mathematics

With the advent of technology, using a calculator for mathematics tasks has become an integral part of problem-solving. Calculators are designed to handle operations and functions that might be time-consuming or error-prone when done by hand. This includes calculations like logarithms, trigonometric functions, and exponential equations.

When using a calculator, it's important to be familiar with your device's functions. For logarithms, particularly, make sure you know whether your calculator defaults to base 10 or another base. Most scientific calculators default to base 10 when you press the 'log' button.

Understanding the order of operations and how to input complex expressions, like fractions or nested parentheses, is also vital. For the given example, placing brackets around the fraction ensures the calculator interprets it correctly before applying the logarithm function. Such meticulous input yields accurate results and teaches students the importance of precision in mathematical computations.