Problem 27 - 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 \(a \neq 1\). $$\log _{a} \sqrt{\frac{x^{6}}{y^{3} z^{5}}}$$

Step-by-Step Solution

Answer

The logarithmic expression \(\log _{a} \sqrt{\frac{x^{6}}{y^{3} z^{5}}}\) can be represented as a sum and difference of logarithms like so: \(3 \log_{a}{x} - \frac{3}{2} \log_{a}{y} - \frac{5}{2} \log_{a}{z}\)

Step 1: Express the radical as a fractional exponent

First, the radical can be expressed as a fractional exponent: \(\log _{a} \left(\frac{x^{6}}{y^{3} z^{5}}\right)^{1/2}\)

Step 2: Apply power rule of logarithms

The power rule of logarithms allows us to remove the exponent, bringing it to front. Therefore: \(\frac{1}{2} \log _{a} \frac{x^{6}}{y^{3} z^{5}}\)

Step 3: Apply quotient rule of logarithms

The quotient rule of logarithms allows us to separate the logarithms with a subtraction operation: \(\frac{1}{2}\left(\log_{a}{x^{6}} - \log_{a}{y^{3}z^{5}}\right)\)

Step 4: Apply product rule of logarithms

Next, the product rule of logarithms allows to separate the terms with a plus operation: \(\frac{1}{2} (\log_{a}{x^{6}} - \log_{a}{y^{3}} - \log_{a}{z^{5}})\)

Step 5: Apply power rule of logarithms again

We can apply the power rule of logarithms to bring down the exponents as coefficients, effectively eliminating them: \(\frac{1}{2} (6 \log_{a}{x} - 3 \log_{a}{y} - 5 \log_{a}{z})\)

Step 6: Distribute the coefficient

Distribute the coefficient \(\frac{1}{2}\) to each term inside the parenthesis: \(3 \log_{a}{x} - \frac{3}{2} \log_{a}{y} - \frac{5}{2} \log_{a}{z}\)

Key Concepts

Power Rule of Logarithms

The power rule of logarithms is a fundamental concept that simplifies expressions involving logarithms with exponents. It states that if you have a logarithm of an expression raised to a power, you can bring the exponent in front of the logarithm as a coefficient. This becomes very helpful for simplifying complex logarithmic expressions. For example:\[\log_b(x^n) = n \cdot \log_b(x)\]When you encounter a situation where an expression within a logarithm is raised to a power, simply move the exponent out of the logarithm and use it as a multiplier.

• This technique makes the expression much easier to work with and understand.
• Remember, this can be applied regardless of whether the expression is a single value or a complex term.

In our original exercise, this rule played a crucial role. After transforming the square root into a fractional exponent, the power rule allowed us to simplify it to form expressions like \[\log_a\left(\left(\frac{x^6}{y^3z^5}\right)^{1/2}\right)\] which then turn into \[\frac{1}{2}\log_a\left(\frac{x^6}{y^3z^5}\right)\]. This step reduces complexity significantly.

Quotient Rule of Logarithms

The quotient rule of logarithms is an essential tool when dealing with division within logarithmic expressions. It allows you to split a logarithm of a quotient into individual logarithms. According to the quotient rule:\[\log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y)\]By applying this rule, you can transform a single logarithmic fraction into separate logarithms, making them more manageable.

• This transformation is handy to deal with complex terms efficiently.
• Be sure to maintain the order, as subtraction is not commutative.

In our exercise, the rule was used to break down the expression from \[\frac{1}{2} \log_a\left(\frac{x^6}{y^3z^5}\right)\] into \[\frac{1}{2}\left(\log_a(x^6) - \log_a(y^3z^5)\right)\]. This step simplifies complex expressions by separating the numerator from the denominator.

Product Rule of Logarithms

The product rule of logarithms helps manage multiplication within logarithmic expressions by allowing us to separate a logarithm of a product into the sum of individual logarithms. The rule states:\[\log_b(x \cdot y) = \log_b(x) + \log_b(y)\]Using the product rule makes the evaluation and simplification of logarithmic expressions much more straightforward.

• Helpful for simplifying large or multi-variable expressions.
• Like addition, it maintains the associative property.

In the process of solving our initial exercise, this rule was applied to further break down the developed terms from \[\frac{1}{2}\left(\log_a(x^6) - \log_a(y^3z^5)\right)\], resulting in the more manageable \[\frac{1}{2}(\log_a(x^6) - \log_a(y^3) - \log_a(z^5))\]. Each part can be individually assessed for easier computation.