Problem 85 - Elementary and Intermediate Algebra

Question

Write logarithmic expression as one logarithm. \(\frac{1}{2} \log _{6}\left(x^{2}+1\right)-\log _{6}\left(x^{2}+2\right)\)

Step-by-Step Solution

Answer

The expression simplifies to \(\log _{6}\left(\frac{\sqrt{x^{2}+1}}{x^{2}+2}\right)\).

Step 1: Apply the Power Rule

The power rule for logarithms states that for any logarithm \( ext{log}_b(a^n) = n\log_b(a)\). Here, we have \(\frac{1}{2} \log _{6}(x^{2}+1)\). Apply the power rule in reverse, converting \(\frac{1}{2} \log _{6}(x^{2}+1)\) to \(\log _{6}((x^{2}+1)^{1/2})\). Simplifying gives \(\log _{6}(\sqrt{x^{2}+1})\).

Step 2: Apply the Quotient Rule

The quotient rule for logarithms states \(\log_b(a) - \log_b(c) = \log_b\left(\frac{a}{c}\right)\). We have \(\log _{6}(\sqrt{x^{2}+1}) - \log _{6}(x^{2}+2)\). Apply the quotient rule to combine these into one logarithm: \(\log _{6}\left(\frac{\sqrt{x^{2}+1}}{x^{2}+2}\right)\).

Step 3: Final Solution Check

Our single logarithmic expression is \(\log _{6}\left(\frac{\sqrt{x^{2}+1}}{x^{2}+2}\right)\). Verify the steps: first apply the power rule, then apply the quotient rule correctly. Examining the components ensures all transformations align with logarithmic identities.

Key Concepts

Understanding the Power Rule in Logarithms

The power rule is a handy tool when working with logarithms. It allows you to move an exponent in front of the logarithm and vice versa.
This is particularly useful when you have something multiplied to a logarithm expression.
The rule states that \ \\[ \log_b(a^n) = n \log_b(a) \]
Where \( b \) is the base, \( a \) is the argument, and \( n \) is the exponent.
The reverse process is also true: you can convert back by moving the multiplier to the exponent.
In the given exercise, there is a multiplication of \( \frac{1}{2} \) with the logarithm: \( \frac{1}{2} \log _{6}(x^{2}+1) \).
By using the power rule, we rewrite this as:

• \( \log _{6}((x^{2}+1)^{\frac{1}{2}}) \)


• Which simplifies to \( \log _{6}(\sqrt{x^{2}+1}) \)

This helps in transforming the expression into a more usable form for further operations, like applying the quotient rule next.

Exploring the Quotient Rule

The quotient rule is another crucial identity in logarithms.
It's used when you subtract two logs with the same base.
The rule is expressed as: \ \\[ \log_b(a) - \log_b(c) = \log_b\left(\frac{a}{c}\right) \]This mathematical property helps to combine two logarithms into one.
In our exercise, we are given:

• \( \log _{6}(\sqrt{x^{2}+1}) - \log _{6}(x^{2}+2) \)

Using the quotient rule, this expression becomes a single logarithm:

• \( \log _{6}\left(\frac{\sqrt{x^{2}+1}}{x^{2}+2}\right) \)

This conversion is very useful to simplify expressions down.
The result is a cleaner, more manageable single logarithm.

Logarithmic Identities: A Brief Overview

Logarithmic identities form the foundation of manipulating logs effectively.
They are foundational rules that include the power rule and quotient rule, which we've already explored.
Understanding these identities ensures you can simplify logarithmic expressions methodically and accurately. Here are some important logarithmic identities:

• Product Rule: \( \log_b(a \cdot c) = \log_b(a) + \log_b(c) \)


• Quotient Rule: \( \log_b(a) - \log_b(c) = \log_b\left(\frac{a}{c}\right) \)


• Power Rule: \( \log_b(a^n) = n \log_b(a) \)

These identities are valuable tools.
They enable transformations that may seem complex at first.
Applying these consistently will make understanding and simplifying logarithmic expressions easier over time.