Problem 27 - Algebra 2

Question

Expand each logarithm. \(\log \sqrt{\frac{2 x}{y}}\)

Step-by-Step Solution

Answer

The expanded form of the given logarithm \(\log \sqrt{\frac{2 x}{y}}\) is \(\frac{1}{2}\log 2 + \frac{1}{2}\log x - \frac{1}{2}\log y\).

Step: Express the square root as a power of 1/2

This is the first crucial step that is needed to simplify the expression. So, express the square root as a power of 1/2; we have, \(\log \left(\frac{2 x}{y}\right)^{\frac{1}{2}}\).

Step: Apply the power rule of the logarithm

By using the rule of logarithm stated above, \(m \log_a b\), where m in this case equals to 1/2 (the square root), gives us: \(\frac{1}{2}\log \left(\frac{2 x}{y}\right)\).

Step: Divide the logarithm

Another logarithmic rule states that the division inside the logarithm \(\frac{a}{b}\) can be expressed as \(\log a - \log b\). Applying this rule to \(\frac{1}{2}\log \left(\frac{2 x}{y}\right)\), rewrites the expression as: \(\frac{1}{2}(\log 2x - \log y)\)

Step: Apply the multiplication rule

Now apply the third property of logarithms, the multiplication rule which states: \(\log a*b = \log a + \log b\). Apply this to \(\frac{1}{2}(\log 2x - \log y)\) and we get: \(\frac{1}{2}(\log 2 + \log x - \log y)\)

Step: Distribute the 1/2 factor

The final step is to distribute the factor \(\frac{1}{2}\) to each term inside the parenthesis. This leads us to: \(\frac{1}{2}\log 2 + \frac{1}{2}\log x - \frac{1}{2}\log y\). This step completes the expansion of the logarithm.

Key Concepts

Logarithm Properties

Logarithms may seem intimidating at first, but they follow some simple rules much like basic arithmetic. The properties of logarithms can make complex calculations more manageable by transforming multiplication into addition, division into subtraction, and exponentiation into multiplication. These core properties include:

• Product Rule: Allows you to separate multiplication inside a logarithm as a sum of individual logs.
• Quotient Rule: Transforms division inside a logarithm into a difference of logs.
• Power Rule: Lets you pull the exponent out of the log as a multiplier.

Understanding and applying these properties can significantly simplify expressions. Learning these rules is like learning new computational shortcuts, which are very handy in various mathematical and real-world problems.
By becoming comfortable with these properties, you'll be able to expand and simplify logarithmic expressions with ease.

Power Rule of Logarithms

The Power Rule of logarithms is a valuable tool when dealing with exponents inside a logarithmic expression. Essentially, this rule states that the log of a number raised to an exponent can be expressed as the exponent multiplying the log of the number itself. Mathematically, this is written as:

• \[\log_b(x^n) = n \cdot \log_b(x)\]

This means you can take any exponent outside of the log, simplifying the expression.
For example, in the original exercise, we first expressed the square root as a power: \(\sqrt{value} = value^{1/2}\).
The next step was applying the Power Rule. The log of our expression became a fraction multiplying the log, which significantly simplified the rest of the calculation.
Understanding and using the Power Rule efficiently makes dealing with exponents within logarithms straightforward and less error-prone.

Logarithm Division Rule

The Logarithm Division Rule is essential when you have a division inside a logarithm. Instead of calculating it directly, the Division Rule allows you to break it down into subtraction.
The rule is simple: the log of a quotient is the log of the numerator minus the log of the denominator. It's presented as:

• \[\log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y)\]

In our exercise, this rule helped us transform \(\frac{1}{2}\log\left(\frac{2x}{y}\right)\) into \(\frac{1}{2}(\log 2x - \log y)\).
This conversion lightens our calculation load by substituting division with simple subtraction, making it straightforward to handle and expand the log expression efficiently.
By mastering this rule, you'll soon find reduction from complex expressions to manageable calculations even in more complicated problems.

Logarithm Multiplication Rule

When factors are multiplied inside a logarithm, the Multiplication Rule allows us to split this into a sum of logarithms. This separation makes large, unwieldy expressions easier to manage and this rule states:

• \[\log_b(a \cdot b) = \log_b(a) + \log_b(b)\]

In our context, this rule transformed \(\log 2x\) into \(\log 2 + \log x\).
This step-by-step splitting of components can then be manipulated further, such as deploying the Power Rule as needed.
The benefit of using the Multiplication Rule is evident in simplifying complex multiplicative relationships within logs, thus converting potentially compounded arithmetic into simple additions, facilitating a clearer step-by-step expansion of the logarithmic expressions.