Problem 17 - 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. $$4 \log 2+3 \log 3-2 \log 4$$

Step-by-Step Solution

Answer

\(\log(27)\)

Step 1: Apply Power Rule of Logarithms

Use the power rule, which states that the logarithm of a number raised to a power is equal to the power multiplied by the logarithm of the number. Convert the coefficients into exponents. So, express each term as \(4 \log 2 = \log(2^4)\), \(3 \log 3 = \log(3^3)\), and \(2 \log 4 = \log(4^2)\).

Step 2: Simplify each logarithmic term

Calculate the exponents for each term: \(2^4 = 16\), \(3^3 = 27\), and \(4^2 = 16\). Rewrite the terms using these values: \(\log(16)\), \(\log(27)\), and \(\log(16)\).

Step 3: Use Logarithmic Properties to Combine Terms

Apply the logarithmic properties that \(\log(a) + \log(b) = \log(ab)\) and \(\log(a) - \log(b) = \log(a/b)\) to combine the terms into a single logarithm. Write as \(\log(16) + \log(27) - \log(16)\) which simplifies to \(\log(16 \cdot 27) - \log(16)\) or \(\log(432) - \log(16)\).

Step 4: Subtract Logarithms with Same Base

Since the base of the logarithms is the same and is not explicitly given, it's assumed to be the same (base 10 by convention). Use the subtraction property \(\log(a) - \log(b) = \log(a/b)\) and write the result as \(\log(432/16)\).

Step 5: Simplify the Quotient Inside the Logarithm

Divide 432 by 16 to get \(\log(27)\), since \(432 / 16 = 27\). The expression is now a single logarithm with a coefficient of 1.

Key Concepts

Power Rule of Logarithms

The power rule of logarithms is a transformation technique that can greatly simplify solving and understanding logarithmic expressions. This rule states that when you have an exponent inside a logarithm, such as \( \log(b^n) \) you can move the exponent to the outside, multiplying it by the logarithm. In other words, this power rule is expressed mathematically as \( \log(b^n) = n \cdot \log(b) \).

Applying this rule makes it easier to manipulate logarithmic terms when they are part of a larger expression. For instance, in our given exercise \(4 \log 2\), we can apply the power rule by bringing the coefficient 4 into the logarithm as an exponent, converting it into \(\log(2^4)\). This is not just an algebraic trick—it's rooted in the way logarithms work and how they interact with exponents, reflecting the inverse relationship logarithms have with exponentiation.

Utilizing this property helps in reducing complex logarithmic expressions to a more manageable form. It's especially helpful in the context of combining logarithms or solving logarithmic equations where exponents are involved.

Combining Logarithms

Combining logarithms is another technique frequently used to simplify logarithmic expressions. To effectively use this method, it's crucial to understand two main properties: the product rule and the quotient rule of logarithms. The product rule states that the logarithm of a product is the sum of the logarithms \( \log(a \cdot b) = \log(a) + \log(b) \) and the quotient rule states that the logarithm of a quotient is the difference of the logarithms \( \log(a / b) = \log(a) - \log(b) \).

In our example, after applying the power rule to get expressions like \( \log(16) + \log(27) - \log(16) \), we can combine them using these logarithm properties. This strategy is evident in the step where \( \log(432) - \log(16) \) is further simplified by using the quotient rule, resulting in \( \log(432/16) \). It’s like reconstructing a puzzle by fitting all pieces (logarithmic terms) into one compact form, a single logarithm.

Ultimately, understanding how to combine logarithms not only aids in solving equations but also provides insight into the relationships between numbers and the concepts of multiplication and division within the logarithmic context.

Logarithmic Expressions

Logarithmic expressions are more than just mathematical symbols; they represent the power to which a base must be raised to produce a given number. Understanding how to evaluate and simplify these expressions is integral to solving problems in algebra and beyond.

When dealing with logarithmic expressions, remember that they follow their own set of rules, different from arithmetic and algebra. These rules, such as the power rule, product rule, and quotient rule, as demonstrated in our exercise, are essential for breaking down complex logarithmic equations into simpler parts. By mastering the use of these properties, one gains the ability to rewrite expressions in ways that reveal the underlying relationships and make calculations feasible.

In the context of our problem, transforming \( \log(432) - \log(16) \) into \( \log(27) \) is a perfect example of simplifying a logarithmic expression. This kind of manipulation is at the heart of working with logarithms, whether as part of academic coursework or while tackling real-world phenomena involving exponential growth, such as compound interest or population dynamics. Effective manipulation of logarithmic expressions enables us to solve equations that would otherwise be too complex or unmanageable.