Problem 25 - Precalculus : Building Concepts and Connections

Question

In Exercises \(21-30,\) write each logarithm as a sum and\or difference of logarithmic expressions. Eliminate exponents and radicals and evaluate logarithms wherever possible. Assume that \(a, x, y\) \(z>0\) and \(a \neq 1\). $$\log _{a} \frac{\sqrt{x^{2}+y}}{a^{3}}$$

Step-by-Step Solution

Answer

The final simplified form of the original expression is \( \log_a (x^2 + y)^{1/2} - 3 \)

Step 1: Apply the Quotient Law of Logarithms

Using the quotient rule, the given expression can be simplified as \(\log _{a} \sqrt{x^{2}+y} - \log _{a} a^{3}\)

Step 2: Apply the Power Rule and Root Assignment in Logarithms

The power rule of logarithms helps simplify the second part of the equation and the square root is expressed as a power of 1/2, giving \( \log _{a} (x^{2}+y)^{1/2} - 3 \log _{a} a \)

Step 3: Evaluate Logarithms

We know that \( \log_a a = 1 \), and thus \( \log_a a^3 = 3 \cdot 1 = 3 \). So, the simplified expression would be \( \log_a (x^2 + y)^{1/2} - 3 \)

Key Concepts

Understanding the Quotient Rule

The quotient rule in logarithms is an important concept when simplifying expressions involving division.
It states that the logarithm of a quotient is equal to the difference between the logarithms of the numerator and the denominator.
Mathematically, this rule can be written as:

• \( \log_b \frac{M}{N} = \log_b M - \log_b N \)

This principle is extremely useful in breaking down complex expressions into simpler form.
In the exercise, applying the quotient rule allowed us to separate \( \log_a \frac{\sqrt{x^2+y}}{a^3} \) into two parts:

• \( \log_a \sqrt{x^2+y} \)
• - \( \log_a a^3 \)

It's important to note that the base of the logarithms remains consistent throughout.
This simplification is foundational in solving logarithmic equations more efficiently.

Exploring the Power Rule

Next comes the power rule, another key tool in simplifying logarithmic expressions.
This rule states that when you have a logarithm of a number raised to a power, you can bring the exponent down as a multiplier.

• The power rule is expressed as: \( \log_b M^n = n \cdot \log_b M \)

In the problem at hand, the expression \( \log_a a^3 \) involves an exponent, which is \( 3 \).
By applying the power rule, we transform it into \( 3 \cdot \log_a a \).
Recognizing \( \log_a a = 1 \) simplifies this further to just \( 3 \).
Additionally, applying the power rule to roots, we express a square root as a power of \( 1/2 \).

• This turns the expression \( \log_a \sqrt{x^2+y} \) into \( \log_a (x^2+y)^{1/2} \).

The power rule is indispensable when simplifying and evaluating logarithmic expressions, especially those involving exponents.

Simplifying Radicals in Logarithms

Radicals, like square roots, can complicate expressions if left in their root form.
To simplify expressions involving radicals inside a logarithm, we express them as fractional powers.

• For instance, \( \sqrt{x} \) can be rewritten as \( x^{1/2} \).

In the given exercise, \( \sqrt{x^2+y} \) was expressed as \( (x^2+y)^{1/2} \).
This allows for further application of the power rule, bringing the expression within the logarithm out as a multiplier:

• \( \log_b x^{1/2} = \frac{1}{2} \cdot \log_b x \)

This method effectively removes the radical by converting it into a manageable exponent.
Understanding how radicals interact with logarithms is the key to simplifying complex logarithmic expressions efficiently.
This also lays the groundwork for further operations like adding or subtracting logarithms.