Problem 47 - Precalculus with Limits

Question

Expanding a Logarithmic Expression In Exercises \(37-58\) , use the properties of logarithms to expand the expression as a sum, difference, and or constant multiple of logarithms. (Assume all variables are positive.) $$\ln z(z-1)^{2}, z>1$$

Step-by-Step Solution

Answer

The expanded form of the expression \(\ln z(z-1)^{2}\) is \(\ln z + 2*\ln(z-1)\).

Step 1: Identify the terms to be expanded

The given expression is \(\ln z(z-1)^{2}\). This expression contains two terms that are multiplied together: \(z\) and \((z-1)^2\).

Step 2: Apply logarithmic properties of products and powers

We can use the logarithmic property of products \(\log(ab) = \log a + \log b\) to simplify the expression. This gives us: \(\ln z + \ln (z-1)^{2}\).

Step 3: Apply the power rule to the second term

Now apply the logarithmic property of powers (\(\log a^{b} = b * \log a\)) to the second term in the sum, resulting in: \(\ln z + 2*\ln(z-1)\).

Key Concepts

Logarithmic Properties

Understanding logarithmic properties is fundamental when it comes to simplifying expressions with logs. A logarithm, in essence, is the inverse of exponentiation and its properties reflect that relationship.

• Product Property: The logarithm of a product is the sum of the logarithms of the individual factors, expressed as \(\log(ab) = \log a + \log b\).
• Quotient Property: Similarly, the logarithm of a quotient is the difference between the logarithm of the numerator and the denominator, written as \(\log(\frac{a}{b}) = \log a - \log b\).
• Power Property: The logarithm of a power is the exponent times the logarithm of the base, given by \(\log a^{b} = b \cdot \log a\).

When expanding logarithmic expressions, these properties allow us to break down complex logs into simpler parts, which can be more easily understood and calculated. For example, when we are given an expression like \(\ln z(z-1)^{2}\), knowing these properties helps us to split it into the sum of individual logarithms and apply the power property, making the expression easier to work with.

Logarithms

Logarithms are not just abstract mathematical concepts; they have practical applications in fields such as science, engineering, and finance. A logarithm answers the question: to what exponent must we raise a certain number (known as the base) to obtain another number? For instance, \(\log_b a = c\) if and only if \(b^c = a\).

The two most common logarithm bases are 10 (common logarithm, \(\log\)) and the mathematical constant e (natural logarithm, \(\ln\)). Logs can help us solve equations involving exponential growth or decay and understand changes in orders of magnitude. In the context of an expression like \(\ln z(z-1)^{2}\), the logarithm helps us break down the product into additive components that relate to the rate at which each factor grows.

Natural Logarithm (ln)

The natural logarithm is a specific type of logarithm with the base e, where e is an irrational number approximately equal to 2.71828. This special number e is the base of the natural logarithm because of its unique properties in calculus, particularly in relation to the concept of growth.

The symbol for the natural logarithm is \(\ln\). The natural logarithm of a number is the power to which e must be raised to obtain the number. For the expression \(\ln z(z-1)^{2}\), we focus on the natural logarithm because of the notation \(\ln\). It's the natural logarithm's relationship with exponential functions that makes it particularly valuable in both theoretical mathematics and applications involving continuous growth or decay, like population models or calculating compound interest.