Problem 54 - Precalculus with Limits

Question

Expanding a Logarithmic Expression In Exercises \(37-58,\) use the properties of logarithms to expand the expression as a sum, difference, and or constant multiple of logarithms. (Assume all variables are positive.) $$\log _{2} x^{4} \sqrt{\frac{y}{z^{3}}}$$

Step-by-Step Solution

Answer

The expanded form of the expression \( \log_2 x^{4} \sqrt{y/z^{3}} \) is \( 4 \cdot \log_2x + \frac{1}{2} \cdot \log_2y - \frac{3}{2} \cdot \log_2z \).

Step 1: Identify the parts of the expression that can be separated

The expression given is \( \log_2 x^{4} \sqrt{y/z^{3}} \). Notice this can be rewritten as \( \log_2(x^{4}) + \log_2\sqrt{y/z^{3}} \). This is due to the property that \( \log_a(bc) = \log_a(b) + \log_a(c) \).

Step 2: Resolve the square root

The square root can be written as a power of \( \frac{1}{2} \). So the expression becomes \( \log_2(x^{4}) + \log_2{(y/z^{3})^{1/2}} \).

Step 3: Expand the expression further

Again, applying the rule \( \log_a(b^n) = n \cdot \log_a(b) \), the expression now becomes \( 4 \cdot \log_2x + \frac{1}{2} \cdot \log_2(y/z^{3}) \).

Step 4: Apply the property for division

Identifying that we can consider \( y/z^3 \) as \( y/z^{3} \), we can apply the property \( \log_a(b/c) = \log_a(b) - \log_a(c) \) so, our expression becomes \( 4 \cdot \log_2x + \frac{1}{2}(\log_2y - \log_2z^{3}) \).

Step 5: Simplify the expression

Finally applying \( \log_a(b^n) = n \cdot \log_a(b) \) once more on the last term, the expanded form of the original expression is \( 4 \cdot \log_2x + \frac{1}{2} \cdot \log_2y - \frac{3}{2} \cdot \log_2z \).

Key Concepts

Properties of Logarithms

Understanding the properties of logarithms is essential when working with logarithmic expressions. The logarithm of a product, quotient, or power has properties that allow you to rewrite the expression in a more straightforward form.

Firstly, the product rule states that the logarithm of a product is the sum of the logarithms of the individual factors: \[ \log_a(bc) = \log_a(b) + \log_a(c) \.\] Similarly, the quotient rule allows the division inside a logarithm to be split into the difference of two logarithms: \[ \log_a\left(\frac{b}{c}\right) = \log_a(b) - \log_a(c) \.\] Last but not least, the power rule simplifies the logarithm of a power by bringing the exponent out as a multiplier: \[ \log_a(b^n) = n \cdot \log_a(b) \.
\] By applying these properties systematically, we can expand complex logarithmic expressions into simpler terms that are easier to manipulate and understand.

Logarithmic Expansion

The process of logarithmic expansion leverages the properties of logarithms to rewrite a single logarithmic expression into an equivalent sum, difference, or product of simpler logarithms. It simplifies the understanding and calculation of the logarithmic expression.

As in the provided example, the expansion begins by separating products and quotients within the logarithm using the product and quotient rules. If there are exponents, the power rule is applied, which turns multiplication within the logarithm into multiplication outside of the logarithm. When performed step by step, as shown in the solution, the original expression becomes a linear combination of simpler logarithmic terms. This makes it more accessible, especially when taking derivatives, integrating, or solving equations.

Logarithm Rules

Mastering logarithm rules is a fundamental part of expanding logarithmic expressions. Beyond the product, quotient, and power rules discussed earlier, there are additional rules that are equally important.

For instance, one of the basic rules states that the logarithm of one, regardless of the base, is always zero: \[ \log_a(1) = 0 \.
\] Additionally, the logarithm of the base itself is always one: \[ \log_a(a) = 1 \.\]
Another rule, often referred to as the change of base formula, allows you to convert a logarithm of one base to another, which is especially useful when the base is not common: \[ \log_b(a) = \frac{\log_c(a)}{\log_c(b)} \.
\] By keeping these rules in mind, students can approach logarithmic expressions with confidence, ensuring no step is overlooked during expansion.