Problem 64 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the properties of logarithms to expand the expression as a sum, difference, and/or constant multiple of logarithms. (Assume all variables are positive.)$$\log _{b} \frac{\sqrt{x} y^{4}}{z^{4}}$$.

Step-by-Step Solution

Answer

The expansion of the given expression is \( 0.5\log_{b}x + 4\log_{b}y - 4\log_{b}z \).

Step 1: Apply the Quotient Rule

First, apply the quotient rule which states that the logarithm of a quotient equals the difference of the logarithms. This results in: \( \log_{b}\sqrt{x} y^{4} - \log_{b} z^{4} \).

Step 2: Apply the Product Rule

Next, apply the product rule which tells us that the logarithm of a product equals the sum of the logs of the factors. The expression then becomes: \( \log_{b}\sqrt{x} + \log_{b}y^{4} - \log_{b}z^{4} \).

Step 3: Apply the Power Rule

Lastly, apply the power rule which states that the logarithm of a number raised to an exponent equals the exponent times the logarithm of the base. We also remember that the square root of x is equivalent to x raised to the power of 0.5. This simplifies the expression to: \( 0.5\log_{b}x + 4\log_{b}y - 4\log_{b}z \).

Key Concepts

Quotient Rule

When dealing with logarithms, one useful property is the quotient rule. This rule helps to simplify logarithmic expressions that involve division. According to the quotient rule:

• The logarithm of a quotient is equal to the difference between the logarithms of the numerator and the denominator.

In mathematical terms, this is expressed as:\[\log_{b}\left(\frac{m}{n}\right) = \log_{b}m - \log_{b}n\]
Applying this rule to an expression means you subtract the log of the denominator (the part of the fraction on the bottom) from the log of the numerator (the part on top).
For instance, in our exercise, we have:\[\log_{b} \frac{\sqrt{x} y^{4}}{z^{4}}\]
Applying the quotient rule, it becomes:\[\log_{b}(\sqrt{x} y^{4}) - \log_{b}(z^{4})\]
This sets the stage for further simplifications using other logarithmic rules.

Product Rule

The product rule of logarithms is another handy tool when expanding or simplifying logarithmic expressions. As the name suggests, it is used when the expression involves multiplication.

• The logarithm of a product is equal to the sum of the logarithms of the individual factors.

Mathematically, the rule is written as:\[\log_{b}(m \cdot n) = \log_{b}m + \log_{b}n\]
This means that when you have a multiplication inside a logarithm, you can break it into separate logarithms and add them together.
In the given problem, once we have applied the quotient rule:\[\log_{b}(\sqrt{x} y^{4}) - \log_{b}(z^{4})\]
We then apply the product rule to the term \(\log_{b}(\sqrt{x} y^{4})\):\[\log_{b}(\sqrt{x}) + \log_{b}(y^{4})\]
Thus, the expression is now simplified to:\[\log_{b}(\sqrt{x}) + \log_{b}(y^{4}) - \log_{b}(z^{4})\]
This makes the expression easier to handle for the next step, where the power rule comes into play.

Power Rule

The power rule is incredibly helpful for simplifying logarithmic expressions that contain exponents. This rule lets you take any exponent on a number inside a logarithm and move it out in front as a multiplier:

• The logarithm of a power is the exponent times the logarithm of the base.

The formula looks like this:\[\log_{b}(m^{n}) = n \cdot \log_{b}m\]
For the expression we simplified using both the quotient and product rules:\[\log_{b}(\sqrt{x}) + \log_{b}(y^{4}) - \log_{b}(z^{4})\]
We recognize that \(\sqrt{x}\) is \(x^{0.5}\). Thus:\[\log_{b}(x^{0.5}) = 0.5 \cdot \log_{b}x\]
For \(y^{4}\) and \(z^{4}\), apply the rule as follows:\[\log_{b}(y^{4}) = 4 \cdot \log_{b}y\]\[\log_{b}(z^{4}) = 4 \cdot \log_{b}z\]
Combining these, the expanded form becomes:\[0.5 \log_{b}x + 4 \log_{b}y - 4 \log_{b}z\]
This expansion makes it easier to work with and understand complex logarithmic expressions.