Problem 13 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Given that \(\log 3 \approx 0.4771\) and \(\log 4 \approx\) 0.6021, find the value of each logarithm. $$\log 16$$

Step-by-Step Solution

Answer

Using the given approximations \(\log 3 \approx 0.4771\) and \(\log 4 \approx 0.6021\), we find that \(\log 16\) can be written as \(2 \log 4\). Substituting the value of \(\log 4\), we get \(2 (0.6021) = 1.2042\). Therefore, \(\log 16 \approx 1.2042\).

Step 1: Rewrite the expression

Given that \(16 = 4^2\), we rewrite the desired logarithm as: $$\log 16 = \log (4^2)$$

Step 2: Apply logarithmic properties

To simplify the expression, use the power rule of logarithms: \(\log_a b^c = c \log_a b\). Applying this rule to our expression: $$\log (4^2) = 2 \log 4$$

Step 3: Substitute the approximate logarithm values

Now that we have the expression \(2 \log 4\), we can use the given approximate value of \(\log 4 \approx 0.6021\) to find the approximate value of the original logarithm: $$2 \log 4 \approx 2 (0.6021)$$

Step 4: Calculate the approximate value

Multiply the coefficient by the approximate value: $$2 (0.6021) = 1.2042$$ This gives us the approximate value of \(\log 16\): $$\log 16 \approx 1.2042$$

Key Concepts

Logarithm Power Rule

Understanding the logarithm power rule is fundamental to simplifying logarithmic expressions. This rule states that the logarithm of a number raised to an exponent is equal to the exponent multiplied by the logarithm of the number itself. In mathematical terms,
\[\begin{equation}\log_b(a^c) = c \log_b(a)\end{equation}\]
This is particularly useful because it converts a potentially complex logarithmic calculation into a much simpler multiplication problem. For instance, if you have \[\begin{equation}\log(4^2)\end{equation}\], you can simplify it by using the power rule to \[\begin{equation}2 \log(4)\end{equation}\], as shown in the original exercise. The real beauty of the power rule comes into light when dealing with large exponents, as it breaks down the problem into more manageable parts.

Simplifying Logarithms

When faced with logarithmic expressions, simplifying them makes the numbers easier to work with. Simplification often involves employing various properties of logarithms, such as the power rule. Simplifying can turn an intimidating logarithmic expression into something more familiar and easily calculable.
This is especially true when the logarithms involve approximations or need to be estimated for further calculations. For instance, knowing that \[\begin{equation}\log 4 \approx 0.6021\end{equation}\], allows us to find the approximate value of any logarithm base 10 with a base of 4. When you follow a step-by-step approach, as outlined in the exercise, you can arrive at a precise and clear value without any confusion.

Properties of Logarithms

Logarithms come with a set of powerful properties that allow for the manipulation and simplification of expressions. In addition to the power rule, other key properties include the product rule, \[\begin{equation}\log_b(mn) = \log_b(m) + \log_b(n)\end{equation}\], and the quotient rule, \[\begin{equation}\log_b\left(\frac{m}{n}\right) = \log_b(m) - \log_b(n)\end{equation}\]. Moreover, the base change rule and the identity property further expand our ability to work with logarithms: \[\begin{equation}\log_b(a) = \frac{\log_c(a)}{\log_c(b)}\end{equation}\] and \[\begin{equation}\log_b(b) = 1\end{equation}\], respectively. Mastering these properties provides students with a strong toolkit for tackling a wide range of problems in mathematics, engineering, and sciences.