Problem 28 - Algebra 2 and Trigonometry

Question

Write each expression as a single logarithm. \(\log _{e} x+2 \log _{e} y-2 \log _{e} z\)

Step-by-Step Solution

Answer

\(\log_e \left(\frac{x \times y^2}{z^2}\right)\)

Step 1: Understand the Properties of Logarithms

To combine multiple logarithmic expressions into a single logarithm, recall the logarithmic properties: 1) The product property: \(log_b(M) + log_b(N) = log_b(M \times N)\), 2) The power property: \(log_b(M^n) = n \times log_b(M)\), and 3) The quotient property: \(log_b(M) - log_b(N) = log_b(M/N)\).

Step 2: Apply the Power Property

The expression given is \(log_e x + 2 \log_e y - 2 \log_e z\). First, apply the power property to the terms \(2 \times \log_e y\) and \(-2 \times \log_e z\). This results in \(\log_e y^2\) and \(\log_e z^2\), respectively.

Step 3: Apply the Product and Quotient Properties

Now the expression becomes \(\log_e x + \log_e y^2 - \log_e z^2\). Use the product property on the first two terms: \(\log_e(x \times y^2)\). Then, apply the quotient property on this result and \(\log_e z^2\), which gives \(\log_e\left(\frac{x \times y^2}{z^2}\right)\).

Step 4: Write the Result as a Single Logarithm

After applying the logarithmic properties in the previous steps, the expression \(\log_e x + 2 \log_e y - 2 \log_e z\) is simplified to a single logarithm: \(\log_e \left(\frac{x \times y^2}{z^2}\right)\).

Key Concepts

Logarithmic Properties

Logarithmic expressions are simplified using basic rules known as logarithmic properties. These properties help in transforming complex logarithmic expressions into simpler ones, and are crucial tools in algebra.
The most commonly used properties are:

• Product Property: This states that the logarithm of a product equals the sum of the logarithms. For example, \(\log_b(M imes N) = \log_b(M) + \log_b(N) \).
• Power Property: This says that the logarithm of a number raised to an exponent is the exponent times the logarithm of the number. In mathematical terms, \(\log_b(M^n) = n \times \log_b(M) \).
• Quotient Property: According to this property, the logarithm of a quotient is the difference between the logarithms. It's expressed as \(\log_b\left(\frac{M}{N}\right) = \log_b(M) - \log_b(N) \).

Each of these properties plays an essential role in the simplification of logarithmic expressions.

Product Property

The product property of logarithms is quite intuitive and essential when you're working with expressions involving multiplication inside the log. The property states:

• \(\log_b(M \times N) = \log_b(M) + \log_b(N)\)

For example, if you have two numbers, \(a\) and \(b\), and you want to find the logarithm of their product, you can actually break it down into simpler terms: the log of \(a\) plus the log of \(b\). This property is incredibly useful in the simplification of complex expressions, such as combining logs with the same base into a single logarithm.
When using the product property specifically, ensure that the base of the logs you are trying to combine is the same. This is important for the accurate application of the property. If the bases differ, you can't directly apply the product property.

Power Property

The power property of logarithms is a powerful tool, no pun intended, that helps simplify expressions where a logarithm involves exponents. The mathematical representation is:

• \(\log_b(M^n) = n \times \log_b(M)\)

This property means that if you have a power inside the logarithm, you can "bring down" the exponent in front of the log. For instance, in the expression \(2 \log_e y\), applying the power property results in \(\log_e y^2\).
It is often used to simplify expressions before applying other properties, like the product or quotient properties. Without it, manipulating logarithms that involve powers would be considerably more complex. Note that for the power property to work correctly, the base of the logarithm must match the base in the expression you’re simplifying.

Quotient Property

The quotient property of logarithms provides a straightforward way to deal with division inside log expressions. It follows the rule:

• \(\log_b\left(\frac{M}{N}\right) = \log_b(M) - \log_b(N)\)

This property tells us that the logarithm of a division can be rewritten as the difference of two logarithms. In practice, it is commonly used to separate complex expressions involving division into simpler components.
For the exercise at hand, this property plays a key role in transforming the logarithmic expression \(\log_e(x \times y^2) - \log_e(z^2)\) into \(\log_e\left(\frac{x \times y^2}{z^2}\right)\). This change not only simplifies the log expressions but also combines them into a focus-friendly single logarithm format.Applying the quotient property correctly requires consistency in the logarithm base, which ensures the calculations support a cohesive depiction of the mathematical relationship expressed in the equation.