Problem 43 - 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.) $$\ln \sqrt{z}$$

Step-by-Step Solution

Answer

The expanded form of \(\ln \sqrt{z}\) is \(\(1/2) \ln z\)

Step 1: Convert the square root into exponent

First, we have to express the square root as an exponent. The square root of z can be written as \(z^{1/2}\), thus we can rewrite the given expression \(\ln \sqrt{z}\) as \(\ln z^{1/2}\).

Step 2: Apply the power rule of logarithms

Second, according to the power rule of logarithms which states that the logarithm of a power can be written as the exponent times the logarithm of the base, we can simplify \(\ln z^{1/2}\) as \(\(1/2\) \ln z\). This step completes the expansion of the logarithmic expression.

Key Concepts

Expand Logarithmic Expressions

When dealing with logarithmic expressions, one of the tasks you may encounter is their expansion. This means rewriting a single logarithmic term, often involving products, quotients, or powers, into multiple terms that utilize the sum and difference properties of logarithms. This process helps simplify complex logarithmic phrases and makes calculations easier to manage.

Consider the expression given in the exercise:

• We start with \ln \sqrt{z}.

To expand it, we’ll first rewrite the square root in its exponent form, as \(z^{1/2}\). By doing this, we prepare the expression for applying further logarithmic rules, such as the power rule.

Properly expanding logarithmic expressions helps in solving more complex mathematics problems by breaking them down into simpler, more manageable parts.

Power Rule in Logarithms

The power rule is a fundamental property of logarithms, which states that the logarithm of a power can be rewritten as a product. This is articulated in the formula: \[\log_b(x^n) = n \cdot \log_b(x)\]

• When you have an exponent inside a logarithmic expression, you can "bring down" the exponent as a multiplier in front of the logarithm.

In the context of our exercise, once the square root of \(z\) is rewritten as \(z^{1/2}\), the power rule allows us to transform \(\ln z^{1/2}\) to \((1/2) \ln z\).

This rule simplifies expressions by removing the complexity that comes from powers and turns them into simple multiplication, making it easier to work with logarithms in equations.

Exponent Rules

Exponent rules are vital in mathematics, especially when working with expressions involving squares, cubes, and roots. They define how to handle powers of numbers, which come in handy when simplifying logarithmic expressions as we saw in the exercise.

There are several key exponent rules, but one that directly applies here is:

• Converting square roots to exponents: \(\sqrt{x} = x^{1/2}\).

By applying this conversion, the expression \(\ln \sqrt{z}\) can be rewritten as \(\ln z^{1/2}\).

Understanding exponent rules helps you manage and simplify expressions, whether you're transforming roots into powers as we've done, or handling more complex expressions in different mathematical contexts. Mastery of these rules allows for easier manipulation and simplification of various mathematical equations, especially when logarithms are involved.