Problem 4 - Algebra 2

Question

State the property or properties used to rewrite each expression. \(\log _{6} \sqrt[n]{x^{p}}=\frac{p}{n} \log _{6} x\)

Step-by-Step Solution

Answer

The property used to rewrite the logarithmic expression is the Power Rule.

Step 1: Dissect the Expression

Start by looking at the initial expression, \(\log _{6} \sqrt[n]{x^{p}}\). Here, \(x^{p}\) is the base of the root, and n is the degree of the root.

Step 2: Identify the Power Rule

It is known that when a logarithm has an exponent, it can be rewritten by bringing down the exponent in front of the logarithm. This is called the power rule. In the given expression, \(\log _{6} \sqrt[n]{x^{p}}\) can therefore be rewritten as \(\frac{p}{n} \log _{6} x\).

Step 3: Conclusion on Properties

Therefore, the property used to rewrite the expression is the 'Power Rule' of logarithms, which states that \(\log _{b} (a^c) = c \log _{b} a\) for any positive real numbers a and b (with \(b \neq 1\)), and any real number c.

Key Concepts

Power Rule for Logarithms

The power rule for logarithms is a handy property that allows us to simplify expressions that include logarithms with exponents. The rule states:

• If you have a logarithm of a number raised to an exponent, \( \log_{b} (a^c) \), you can move the exponent, \(c\), in front of the logarithm: \( c \log_{b} a \).

This rule is useful because it simplifies expressions, making them easier to work with or compute.
In the exercise, the expression was \( \log_{6} \sqrt[n]{x^{p}} \). Let's break it down.
The expression under the logarithm has an exponent, \(x^{p}\), and it's inside a root, specifically a \(n\)th root which can be rewritten as a fraction, \(\frac{p}{n}\).
Therefore, using the power rule, this expression is simplified to \(\frac{p}{n} \log_{6} x\).
This move turns a complex expression into something more manageable!

Logarithmic Expressions

Understanding logarithmic expressions is key to mastering various algebraic and calculus problems. Logarithms are essentially the inverse of exponential functions and help indicate how many times a certain number, called the base, must be multiplied by itself to arrive at another number.
For example, in the expression \(\log_{b} a\), \(b\) is the base and \(a\) is the number we're interested in.

• A common expression might be \(\log_{10} 100 = 2\), meaning 10 must be squared to get 100.

Logarithmic expressions can be transformed using various properties, like the power rule mentioned previously, which is crucial for simplifying complicated equations.
Understanding this gives you tools to really handle equations that look tough at first glance.

Expressions with Roots

Expressions with roots often look complicated, but when broken down, they become more understandable. A root indicates that we are looking for a number that, when raised to a specific power, gives us the initial number. For instance, a square root is looking for a number that squared gives us our starting number.
In the context of logarithms, expressions within roots can still be simplified using logarithm properties.

• In the exercise, \(\sqrt[n]{x^{p}}\) appears, which is the same as \((x^{p})^{1/n}\).
• When we use the power rule for logarithms, this becomes \(\frac{p}{n} \log_{6} x\).

This transcription from a root expression into a logarithmic format leverages the power rule, making calculation and simplification significantly easier.