Problem 6 - Algebra and Trigonometry with Analytic Geometry

Question

Express in terms of logarithms of \(x, y, z\), or \(w\). $$ \log \frac{\sqrt{y}}{x^{4} \sqrt[3]{z}} $$

Step-by-Step Solution

Answer

\( \frac{1}{2} \log y - 4 \log x - \frac{1}{3} \log z \)

Step 1: Expressing the Fraction as a Difference

Using the property of logarithms that states \( \log \frac{a}{b} = \log a - \log b \), we can write:\[\log \frac{\sqrt{y}}{x^{4} \sqrt[3]{z}} = \log \sqrt{y} - \log (x^{4} \sqrt[3]{z})\]

Step 2: Simplifying the Logarithm of a Power

Using the logarithm power rule \( \log a^{b} = b \cdot \log a \), we simplify:\[\log \sqrt{y} = \log y^{1/2} = \frac{1}{2} \cdot \log y\]

Step 3: Distributing Logarithm Over a Product

Using the property \( \log (a \cdot b) = \log a + \log b \), we apply this to the denominator:\[\log (x^{4} \sqrt[3]{z}) = \log x^{4} + \log \sqrt[3]{z}\]

Step 4: Simplifying Logarithms in Step 3

Continuing to simplify using the power rule:- \( \log x^{4} = 4 \cdot \log x \)- \( \log \sqrt[3]{z} = \log z^{1/3} = \frac{1}{3} \cdot \log z \)

Step 5: Combining All Parts

Substitute the simplified forms back into the equation from Step 1:\[\log \frac{\sqrt{y}}{x^{4} \sqrt[3]{z}} = \frac{1}{2} \cdot \log y - (4 \cdot \log x + \frac{1}{3} \cdot \log z)\]Simplifying further gives:\[\frac{1}{2} \cdot \log y - 4 \cdot \log x - \frac{1}{3} \cdot \log z\]

Key Concepts

Logarithm Properties

Logarithm properties are powerful tools that help simplify complex expressions and make them easier to work with. One fundamental property is the logarithm of a quotient. It states that the logarithm of a division, or fraction, is equal to the difference of the logarithms of the numerator and the denominator. If you have two numbers, say \(a\) and \(b\), the property looks like this: \(\log \frac{a}{b} = \log a - \log b\).
This property allows us to break down a fraction into separate parts, simplifying calculations and transformations. It's crucial in situations where you deal with complex divisions, like in the original exercise, where we turned \(\log \frac{\sqrt{y}}{x^{4} \sqrt[3]{z}}\) into \(\log \sqrt{y} - \log (x^{4} \sqrt[3]{z})\), splitting it into more manageable pieces.
Using these properties helps in various mathematical problems, particularly when you need to rewrite expressions for different purposes or simplify complex logarithmic equations.

Power Rule in Logarithms

The power rule in logarithms is a straightforward yet extremely useful technique to simplify expressions like \(\log a^b\). This rule states that you can take the exponent \(b\) of a logarithmic argument \(a\) and move it in front of the logarithm: \(\log a^b = b \cdot \log a\).
By applying this rule, large exponents become simple multipliers. For instance, transforming \(\log y^{1/2}\) into \(\frac{1}{2} \cdot \log y\) and \(\log x^4\) into \(4 \cdot \log x\) makes them easier to handle and incorporate into broader expressions.
If you've ever found stack of exponential terms overwhelming, the power rule is your best friend, turning potential confusion into clear, simple multiplication. It's particularly useful in logarithmic transformations where direct values are too large or complex to manage easily.

Logarithm of Fractions

When you work with fractions inside a logarithm, there are specific techniques to help you express them in a more usable form. A fraction in a logarithm can mean overwhelming numbers, but with logarithmic properties, these numbers become numbers we can easily manage.
The original equation \(\log \frac{\sqrt{y}}{x^4 \sqrt[3]{z}}\) is a typical example. First, we used logarithm properties to express the fraction by separating the numerator and denominator: \(\log \frac{a}{b} = \log a - \log b\).
Then, we made use of both the power rule, \(\log a^b = b \cdot \log a\), and properties of products, \(\log (a \cdot b) = \log a + \log b\), to further break down and simplify. By recognizing these key relationships, you can solve the fraction issues that arise when working with logarithmic expressions.
This approach, logarithms' flexibility in handling multiplication, division, powers, and roots, makes them an invaluable part of algebra and calculus.