Problem 48 - Precalculus : Building Concepts and Connections

Question

Use the change-of-base formula to evaluate each logarithm using a calculator. Round answers to four decimal places. $$\log _{2} 20$$

Step-by-Step Solution

Answer

The simplified logarithm would be approximately equal to a numerical value, rounded to four decimal places.

Step 1: Applying Change of Base formula

Using the Change of Base Formula, convert the logarithm base 2 to base 10 or base e as per the availability on your calculator. Hence, \(\log _{2} 20 = \frac{\log_{10} 20}{\log_{10} 2}\) or \(\log _{2} 20 = \frac{\ln 20}{\ln 2}\).

Step 2: Calculating the values using a calculator

Calculate the values of \(\log_{10} 20\), \(\log_{10} 2\), or \(\ln 20\), \(\ln 2\) using a scientific calculator and divide the results to get the final answer.

Step 3: Rounding the result

The final step is to round the resulting value to four decimal places. Following standard rounding rules, if the fifth digit is 5 or more, round up the fourth digit by one. If the fifth digit is less than 5, leave the fourth digit as it is.

Key Concepts

Understanding Logarithms

Logarithms are mathematical tools that help us find out how many times a number, called the base, is multiplied by itself to reach another number. For example, in the expression \( \log_2 8 \), we are asking: "How many times must 2 be multiplied by itself to produce 8?" The answer is 3, because \( 2^3 = 8 \). The logarithm gives us the exponent or the power needed for this multiplication.

Knowing how to work with logarithms is essential for solving many problems in mathematics and science. They can simplify complex multiplication into simple addition. Another important use of logarithms is in the measurement of quantities that span many orders of magnitude, like the intensity of an earthquake or the brightness of stars.

In general, the formula \( \log_b x = y \) states that \( b^y = x \), where \( b \) is the base, \( x \) is the number we want to evaluate, and \( y \) is the result. It is important to understand that the base is a crucial part of the logarithm, as it dictates the multiplication scale.

Exploring the Base 10 Logarithm

The base 10 logarithm, often noted as \( \log_{10} \) or simply \( \log \) (especially when the base is understood), is commonly used in many fields such as engineering and science. It is derived from the base 10 system, which is the number system we use in everyday life. This makes it intuitive and efficient for dealing with a wide range of real-world data.

Calculators and computational tools almost always have a dedicated "log" button, which allows for quick computation of base 10 logarithms. This is particularly helpful for converting between multiplicative and additive models, or when comparing quantities spread over a large range of values—like pH levels or sound intensity, which are both measured on a log scale.

To convert a logarithm of an arbitrary base to base 10, the change-of-base formula is used: \( \log_b x = \frac{\log_{10} x}{\log_{10} b} \). This formula allows you to convert any logarithm to a base that can easily be input into most calculators, thereby making your calculations much simpler.

Getting to Know the Natural Logarithm

The natural logarithm, denoted as \( \ln \), is another important type of logarithm that uses the base \( e \), where \( e \approx 2.71828 \). This number is an irrational and transcendental number, meaning it can't be expressed as a fraction and is extremely important in mathematics, particularly in calculus.

Natural logarithms are used to solve growth processes that change at a continuous rate, such as population growth, radioactive decay, or interest compounding. They are essential in calculus because they make differentiation and integration much easier when working with exponential functions.

Just like with base 10 logarithms, you can convert any logarithm to a natural logarithm using the change-of-base formula: \( \log_b x = \frac{\ln x}{\ln b} \). This is especially useful for calculations involving exponential growth or decay, where you're often dealing with the constant \( e \). Calculators provide a direct method to calculate \( \ln \) with buttons specifically for this purpose.