Problem 13 - Technical Mathematics with Calculus

Question

Express as a single logarithm with a coefficient of \(1 .\) Assume that the logarithms in each problem have the same base. $$\log 3+\log 4$$

Step-by-Step Solution

Answer

\log(3) + \log(4) = \log(12)

Step 1: Understand the Properties of Logarithms

Recall the product rule for logarithms, which states that the logarithm of a product is equal to the sum of the logarithms of the factors. In symbolic form, \(\log(a) + \log(b) = \log(ab)\). This property will be used to combine the given logarithms.

Step 2: Combine the Logarithms

Apply the product rule for logarithms to combine \(\log 3\) and \(\log 4\) into a single logarithm. According to the product rule, \(\log 3 + \log 4 = \log(3 \times 4)\).

Step 3: Simplify the Expression

Multiply the numbers inside the logarithm to simplify the expression: \(\log(3 \times 4) = \log(12)\). There is now a single logarithm with a coefficient of 1, as required.

Key Concepts

Product Rule for Logarithms

Understanding the product rule for logarithms is pivotal when working with logarithmic expressions. Simply put, if you have two logarithms with the same base that are being added, such as \(\log(a) + \log(b)\), you can combine them into one by multiplying their inside numbers, giving you \(\log(ab)\).

For example, when you see an expression like \(\log 3 + \log 4\), you can immediately think of it as the logarithm of the product of 3 and 4. This serves as a valuable shortcut to simplify expressions and makes calculating with logarithms much more straightforward.

Logarithm Properties

Logarithms have unique properties that allow mathematicians to manipulate and simplify complex expressions. Besides the product rule, other key properties include the quotient rule, \(\log\frac{a}{b} = \log a - \log b\), and the power rule, \(\log a^b = b \cdot \log a\). These properties are derived from the definitions of logarithms and their relationship to exponents.

Remember, when using these properties, the logarithms must share the same base, as the base represents the number that is being raised to a power to achieve the number inside the logarithm. Mastery of these properties is essential for anyone keen on algebra and higher-level mathematics, as they frequently come up in equations and problem-solving scenarios.

Logarithmic Expressions

A logarithmic expression is a way to represent a power relationship in a different form. Instead of using exponentiation, it tells us what power a base number must be raised to produce another number. But how we handle these expressions isn't just by plugging and chugging numbers; it involves using the aforementioned properties of logarithms to simplify and solve equations.

As you encounter more complex logarithmic expressions, you'll often have to combine multiple properties. For instance, you might need to expand a logarithm using the power rule before you can use the product or quotient rule. With practice and a strong grasp of these properties, you'll be able to navigate through the challenges that logarithms present in mathematics.