Problem 87 - Algebra 2

Question

Expand each logarithm. \(\log _{7} \frac{\sqrt{r+9}}{s^{2} t^{\frac{1}{3}}}\)

Step-by-Step Solution

Answer

\( \frac{1}{2}* \log _{7}(r+9) - 2* \log _{7}(s) - \frac{1}{3}* \log _{7}(t) \)

Step 1: Apply Quotient Rule

Since the logarithm is of a quotient, apply the quotient rule of logarithms here. This states that the \(\log_a(b/c) = \log_a(b) - \log_a(c)\), meaning that the logarithm of a quotient is equal to the logarithm of the numerator subtracted by the logarithm of the denominator. Thus, \(\log _{7} \frac{\sqrt{r+9}}{s^{2} t^{\frac{1}{3}}}\) becomes \(\log _{7} \sqrt{r+9} - \log _{7} s^{2} - \log _{7} t^{\frac{1}{3}}\).

Step 2: Apply the Power Rule

The power rule of logarithms states that \(\log_a(m^p) = p*\log_a(m)\). Apply this rule on each of the terms from Step 1. For the term \(\log _{7} \sqrt{r+9}\), remember that the square root is equivalent to the power of one half, \( \sqrt{r+9} = (r+9)^\frac{1}{2}\). Therefore, this becomes \( \frac{1}{2}* \log _{7}(r+9) \). The other two terms become \( 2* \log _{7}(s) \) and \( \frac{1}{3}* \log _{7}(t) \).

Step 3: Simplify

Merge the results from step two together: \( \frac{1}{2}* \log _{7}(r+9) - 2* \log _{7}(s) - \frac{1}{3}* \log _{7}(t) \).

Key Concepts

Understanding the Quotient Rule for Logarithms

Logarithms are powerful tools in mathematics, especially when dealing with division operations inside the argument of a logarithmic function.
This is where the Quotient Rule comes into play. According to this rule, the logarithm of a quotient is equal to the logarithm of the numerator minus the logarithm of the denominator.
In a formula, it looks like this:

• \(\log_a\left( \frac{b}{c} \right) = \log_a(b) - \log_a(c)\)

In our exercise, we have \(\log_7\left( \frac{\sqrt{r+9}}{s^2 t^{1/3}} \right)\).
Applying the Quotient Rule here separates this into three components:

• \(\log_7\sqrt{r+9}\)
• \(\log_7s^2\)
• \(\log_7t^{1/3}\)

So, it becomes: \(\log_7\sqrt{r+9} - \log_7(s^2) - \log_7(t^{1/3})\).
This transformation greatly simplifies the process of handling division within the logarithmic expression.

Applying the Power Rule with Logarithms

Another key rule when working with logarithms is the Power Rule, which helps when the argument involves powers or roots.
This rule says that the logarithm of a power, like \(m^p\), can be rewritten by multiplying the exponent by the logarithm of the base, \(p\cdot\log_a(m)\).

• This rearranges expressions to better handle calculations.

Let's apply it to the three logarithms from our prior step:

• The term \(\log_7\sqrt{r+9}\) translates to \(\log_7((r+9)^{1/2})\), where the square root is the same as the exponent \(1/2\). Using the Power Rule, this becomes \(\frac{1}{2}\cdot\log_7(r+9)\).
• For \(\log_7(s^2)\), applying the Power Rule results in \(2\cdot\log_7(s)\).
• Similarly, \(\log_7(t^{1/3})\) becomes \(\frac{1}{3}\cdot\log_7(t)\).

These transformations are invaluable for expanding logarithmic expressions and make further arithmetic operations easier.

Mastering Logarithmic Expansion

Logarithmic expansion is a method to deconstruct complex logarithmic expressions into simpler components.
By applying rules like the Quotient and Power rules, we achieve a clear breakdown of the expression. For our problem, the original log expression was:

• \( \log_7\left( \frac{\sqrt{r+9}}{s^2 t^{1/3}} \right) \).

After applying the rules, we reach an expanded form:

• \( \frac{1}{2}\cdot \log_7(r+9) - 2\cdot \log_7(s) - \frac{1}{3}\cdot \log_7(t) \).

This expanded form is beneficial because:

• It facilitates easier integration and differentiation.
• Simplifies evaluation, especially when numerical values are involved.
• Makes it convenient to apply further mathematical transformations.

Understanding and mastering these steps allow you to transform complex logarithms into manageable pieces. Logarithmic expansions are essential tools for tackling advanced mathematical problems.