Problem 29 - 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 \(a \neq 1\). $$\log \sqrt[3]{\frac{x y^{3}}{z^{5}}}$$

Step-by-Step Solution

Answer

\(\frac{1}{3} (\log x + 3 \log y - 5 \log z)\)

Step 1: Recognize the Form of the Logarithmic Expression

The logarithmic expression is in the form \(\log \sqrt[3]{\frac{x y^{3}}{z^{5}}}\). This represents a third root of a fraction, which is equivalent to the fraction raised to the power of \(1/3\). Therefore, the logarithmic expression can be rewritten as \(\log \left(\frac{x y^{3}}{z^{5}}\right)^{1/3}\).

Step 2: Apply the Power Property of Logarithms

Using the power property of logarithms, which states that \(\log b^p = p \log b\), the logarithmic expression can be simplified further to \(\frac{1}{3} \log \left(\frac{x y^{3}}{z^{5}}\right)\).

Step 3: Apply the Quotient Property of Logarithms

Now apply the quotient property of logarithms, which states that \(\log \left( \frac{a}{b} \right) = \log a - \log b\). This simplifies the expression to \(\frac{1}{3} \left( \log xy^{3} - \log z^{5} \right)\).

Step 4: Apply the Product Property of Logarithms

Next, use the product property of logarithms, which states that \(\log ab = \log a + \log b\). This gives \(\frac{1}{3} (\log x + \log y^{3} - \log z^{5})\).

Step 5: Apply the Power Property of Logarithms Again

Finally, you apply the power property of logarithms again to simplify \(\log y^{3}\) to \(3 \log y\) and \(\log z^{5}\) to \(5 \log z\), resulting in the simplified expression \(\frac{1}{3} (\log x + 3 \log y - 5 \log z)\).

Key Concepts

Power Property of Logarithms

The power property of logarithms is instrumental in simplifying logarithmic expressions involving powers. In essence, this property states that if you have a logarithm of a number raised to a power, you can bring the exponent in front of the logarithm for simplification. Mathematically, it is written as:
\[\log_b (a^p) = p \log_b (a)\]Here are some key points to remember:

• The base of the logarithm remains unchanged.
• The exponent becomes a coefficient in front of the logarithm.

Let's use this in practice. Consider the expression \( \frac{1}{3} \log \left(\frac{x y^3}{z^5}\right) \). This can be simplified using the power property by taking the exponent \(1/3\) out of the logarithm, leading to the expression \( \frac{1}{3} \left( \log x + 3 \log y - 5 \log z \right) \).
This step reduces complexity and makes calculations easier, especially when further simplifications or evaluations are necessary.

Quotient Property of Logarithms

The quotient property of logarithms is a super helpful tool when dealing with division inside a logarithm. It states that:
\[\log_b \left( \frac{a}{c} \right) = \log_b a - \log_b c\]This rule splits the logarithm of a quotient into the difference of two logarithms. Let's look at how it can be applied:
Imagine the expression \( \log \left( \frac{x y^3}{z^5} \right) \). By using the quotient property, this can be written as \( \log (x y^3) - \log (z^5) \).

• The numerator becomes one logarithmic term.
• The denominator becomes another, subtracted term.

By applying the quotient property, you simplify the fraction inside the logarithm, making step-by-step calculations more manageable.

Product Property of Logarithms

The product property comes into play when you have a logarithm of a product of terms. The product property of logarithms can be articulated as follows:
\[\log_b (ab) = \log_b a + \log_b b\]This property allows the separation of multiplicative components inside the logarithm into individual sums. Applying this to our example, \( \log (x y^3) \) can be expanded to \( \log x + \log y^3 \).

• Each factor in the product has its own individual logarithm.
• These individual logarithms are then summed together.

Breaking down products in this manner complements subsequent simplifications, especially when these terms are subjected further to other logarithmic properties, such as applying the power property to \( \log y^3 \). Like other properties, it peels apart the components of a logarithmic expression, making it easier to calculate or further manipulate.