Problem 22 - Algebra and Trigonometry Real Mathematics, Real People

Question

Rewrite the expression in terms of \(\ln 4\) and \(\ln 5 .\), $$\ln 500$$.

Step-by-Step Solution

Answer

\(\ln 500 = \ln 4 + 3*\ln 5\)

Step 1: Break down the number

Break down 500 into factors of 4 and 5 i.e., 500 = 4 * 5 * 5 * 5.

Step 2: Apply logarithmic property

Apply the logarithmic property, \(\ln ab = \ln a + \ln b\). For a product of more than two factors, this property expands to \(\ln abc=\ln a+\ln b+\ln c\), and so on. This finally gives \(\ln 500 = \ln 4 + 3*\ln 5\).

Step 3: Simplify the expression

Now the expression \(\ln 500\) is rewritten in terms of \(\ln 4\) and \(\ln 5\).

Key Concepts

Understanding the Natural Logarithm

The natural logarithm, denoted as \( \ln \), is a type of logarithm that uses Euler’s number \( e \) as its base. Euler's number, approximately 2.718, is a fundamental constant in mathematics, often used in growth calculations like compound interest. The natural logarithm answers the question: "To what power must \( e \) be raised, to produce a certain number?" Because of its prevalence in continuous growth processes, the natural logarithm is commonly used in many fields, including physics, economics, and network theory.

When you see an expression like \( \ln x \), it means the power to which \( e \), (2.718), is raised to get \( x \). This property makes natural logarithms powerful for simplifying complex expressions, especially ones involving exponential growth and decay.

The expression \( \ln 500 \) is an example of using properties of the natural logarithm to break down a seemingly complex term into simpler parts.

Factorization for Simplifying Logarithms

Factorization involves breaking a number into a product of simpler integers, known as factors. It's like splitting a whole pie into smaller, manageable slices. In our example, 500 can be broken down into \( 4 \times 5 \times 5 \times 5 \), or written as \( 4 \times 5^3 \). This factorization is the foundation for rewriting expressions in terms of logarithms.

Factorization is useful with logarithms because of a special property: the logarithm of a product equals the sum of the logarithms of the factors.

For example, \( \ln(ab) = \ln a + \ln b \). For a product with multiple terms, like \( \ln(abc) \), it becomes \( \ln a + \ln b + \ln c \).

This ability to break a number into its factors allows us to express complex logarithmic expressions in terms of simpler known logarithms, like \( \ln 4 \) and \( \ln 5 \), as we see in this solution.

Simplifying Expressions Using Logarithmic Properties

Simplifying expressions with logarithms makes complex calculations easier. It involves leveraging key properties to transform unwieldy expressions into more manageable forms. As seen in this exercise, we use the property \( \ln(ab) = \ln a + \ln b \). This allows us to break down a multi-factorial logarithm into a sum of simpler logarithms.

In our step-by-step solution, once we factorized 500 into \( 4 \times 5^3 \), we used the logarithmic product property to write \( \ln 500 \) as \( \ln 4 + \ln 5^3 \).

Further simplification was achieved by using the power rule: \( \ln(a^b) = b \cdot \ln a \). This rule lets us extract exponents out of the logarithm, so \( \ln 5^3 \) becomes \( 3 \cdot \ln 5 \). After applying these rules, the expression simplifies beautifully to \( \ln 4 + 3 \cdot \ln 5 \). This transformation illustrates the power of logarithmic properties in making complex expressions more comprehensible.