Problem 72 - Intermediate Algebra

Question

For Problems \(69-80\), express each of the following as the sum or difference of simpler logarithmic quantities. Assume all variables represent positive real numbers. For example, $$ \begin{aligned} \log _{b} \frac{x^{3}}{y^{2}} &=\log _{b} x^{3}-\log _{b} y^{2} \\ &=3 \log _{b} x-2 \log _{b} y \end{aligned} $$ $$ \log _{b}\left(\frac{x^{2}}{y}\right) $$

Step-by-Step Solution

Answer

\(\log_{b}\left(\frac{x^{2}}{y}\right) = 2\log_{b}x - \log_{b}y\).

Step 1: Identify the Logarithmic Expression

The problem has given us the logarithmic expression \(\log _{b}\left(\frac{x^{2}}{y}\right)\) and we need to express this as a sum or difference of simpler logarithmic terms.

Step 2: Apply the Quotient Rule of Logarithms

According to the quotient rule of logarithms, \(\log_{b}\left(\frac{m}{n}\right) = \log_{b}m - \log_{b}n\). Applying this to \(\log _{b}\left(\frac{x^{2}}{y}\right)\), we can split it into \(\log _{b}x^{2} - \log _{b}y\).

Step 3: Apply the Power Rule to the First Logarithm

The power rule of logarithms says that \(\log_{b}m^{p} = p\log_{b}m\). Thus, \(\log _{b}x^{2}\) can be rewritten as \(2\log _{b}x\).

Step 4: Write the Final Expression

Substitute the simplified first logarithm back into the expression from the quotient rule application. The expression becomes \(2\log _{b}x - \log _{b}y\).

Key Concepts

Quotient Rule of Logarithms

One of the fundamental rules of logarithms is the **Quotient Rule**, which helps simplify logarithmic expressions involving divisions. The quotient rule states: \[\log_{b}\left(\frac{m}{n}\right) = \log_{b}m - \log_{b}n\] This means that when you have a logarithm of a fraction, you can split it into the difference of two logarithms. The numerator of the fraction becomes the first logarithm, and the denominator becomes the second logarithm, with a minus sign between them.
In our problem, we began with \(\log_{b}\left(\frac{x^2}{y}\right)\). Applying the quotient rule, we decompose it into simpler expressions:

• \(\log_{b}x^2\) corresponding to the numerator \(x^2\)
• \(\log_{b}y\) corresponding to the denominator \(y\)

Thus, we break it down to \(\log_{b}x^2 - \log_{b}y\). This step is crucial for simplifying complex logarithmic fractions into more manageable parts.

Power Rule of Logarithms

To take this simplification even further, the **Power Rule** of logarithms comes into play. This rule states: \[\log_{b}m^{p} = p\cdot\log_{b}m\] This means when you have a logarithm of a number raised to a power, you can bring the power in front of the logarithm as a multiplier. It makes expressions more straightforward and easier to understand.
Returning to our example, we have \(\log_{b}x^2\). Using the power rule, we transform it:

• The exponent \(2\) comes down as a coefficient.
• Our expression becomes \(2\cdot\log_{b}x\).

By adjusting the expression this way, we simplify the logarithm without changing its value, turning complicated expressions into simple, easy-to-handle forms.
This transformation from the power rule is essential for dealing with powers present in logarithmic terms, allowing you to express them as a product which can be easier to work with mathematically.

Simplifying Logarithmic Expressions

Ultimately, the goal of using logarithmic properties is to **simplify logarithmic expressions** as much as possible, making them easier to manipulate and understand. In our example, we applied two fundamental rules: the quotient rule and the power rule.
Starting with \(\log_{b}\left(\frac{x^2}{y}\right)\), the quotient rule broke the fraction into the difference of two logarithms, \(\log_{b}x^2 - \log_{b}y\). Then, the power rule simplified \(\log_{b}x^2\) into \(2\log_{b}x\).

• Final Expression: \(2\log_{b}x - \log_{b}y\)
• This means we've simplified the complex original expression to a more straightforward form.

This breakdown is crucial for solving more advanced mathematical problems where handling simpler expressions is necessary for further computation.
By understanding these properties and how they apply, you can tackle complex logarithmic problems efficiently. The goal is to reduce expressions to their simplest forms, making subsequent mathematical operations much more manageable.