Question

Use a calculator to evaluate the logarithm. Round your result to three decimal places.\(\log _{10} 163\)

Short answer

The value of \(\log _{10} 163\) rounded to three decimal places is 2.212

Step-by-step solution

Understand the Logarithm

The logarithm requested here is base 10, also known as the common logarithm. Remember that the logarithm is the inverse function to exponentiation. So, \(\log _{10} a = b\) is equivalent to \(10^b = a\). Here, we want to find \(b\) in \(\log _{10} 163 = b\) which is the same as \(10^b = 163\).

Use a Calculator

This exercise asks for the use of a calculator. Using a calculator, input 'log10(163)' into the calculator and it should return a number approximately equal to 2.212

Round your Result

The number obtained from the calculator will likely have many decimal places. The task however requires the result to be rounded off to three decimal places. Paying attention to the fourth decimal digit (which is 2), one can conclude that the third digit remains unchanged. Thus, the answer is 2.212.

Key concepts

Logarithms and Exponentiation

When students encounter the task of calculating logarithms, understanding the relationship between logarithms and exponentiation is the key to grasping the concept. A logarithm answers the question: 'To what exponent must we raise a certain base to obtain a given number?' Specifically, a common logarithm, with a base of 10, can be represented as \(\log _{10} x = y\), which is the inverse of the expression \(10^y = x\).

For instance, if we want to find the logarithm of 163 base 10, we are essentially looking for the exponent \( y \) that makes the equation \(10^y = 163\) true. The beauty of logarithms in mathematics is their ability to simplify the operation of exponentiation, especially when dealing with large numbers or complex calculations. By transforming multiplicative relationships into additive ones, logarithms are a cornerstone in various areas of math and science.

Logarithmic Function Properties

The properties of logarithmic functions are significant tools that facilitate their calculation and understanding. Some properties to keep in mind include the fact that the logarithm of 1 to any base is always 0 because any number to the power of 0 is 1 \(\log_b(1) = 0\). Another pivotal property is that the logarithm of a base to its own number is always 1 because a number to the power of 1 is itself \(\log_b(b) = 1\).

Product and Quotient Rules

Logarithms exhibit specific behaviors when dealing with multiplication or division. The logarithm of a product is equal to the sum of the logarithms of the individual numbers \(\log_b(xy) = \log_b(x) + \log_b(y)\), whereas the logarithm of a quotient is the difference between the logarithm of the numerator and the denominator \(\log_b(\frac{x}{y}) = \log_b(x) - \log_b(y)\).

Power Rule

Furthermore, when an argument is raised to a power, the exponent can be brought to the front as a multiplier \(\log_b(x^n) = n\log_b(x)\). These properties not only make calculations more straightforward but also enable one to solve complex logarithmic equations and to understand their behavior within different mathematical contexts.

Scientific Calculator Usage

Utilizing a scientific calculator efficiently is a fundamental skill for solving logarithms, among other mathematical operations. To evaluate a common logarithm like \(\log _{10} 163\), a scientific calculator is indispensable. Students should ensure their calculator is in the correct mode for logarithmic functions.

To find a common logarithm, typically, you would press the 'log' button followed by the number whose logarithm you wish to calculate. In this case, you would enter 'log' then '163' and hit the equals key. The calculator will then display the logarithmic value, which can be rounded as necessary.

For logs with bases other than 10 or the natural logarithm base \(e\), most scientific calculators provide the option to compute logarithms with different bases or use the change of base property: \(\log_b(a) = \frac{\log_c(a)}{\log_c(b)}\), where \(c\) is a base that your calculator can handle (usually 10 or \(e\)). When rounding your result, always take care to check the subsequent digit to determine if the last digit in your answer should be rounded up or remain the same. Accurate use of a scientific calculator is essential for achieving precise results in mathematics.