Problem 256 - Precalculus

Question

For the following exercises, expand each logarititim as much as possible. Rewite each expression as a sum, difference, or product of logs. $$ \log _{2}\left(y^{x}\right) $$

Step-by-Step Solution

Answer

\( \log_{2}(y^x) = x \cdot \log_{2}(y) \).

Step 1: Apply the Power Rule of Logarithms

The power rule of logarithms states that for any logarithm \( \log_b(a^c) \), it can be expanded as \( c \cdot \log_b(a) \). In this exercise, identify that \( y^x \) is in the form of \( a^c \), so you can apply the power rule:\[ \log_{2}(y^x) = x \cdot \log_{2}(y) \].

Step 2: Check for Further Expansion

Upon initial expansion, check if the expression can be expanded further. Since \( \log_{2}(y) \) cannot be expanded further without additional information about \( y \), the expression is fully expanded.

Key Concepts

Power Rule of Logarithms

The power rule is a key tool in simplifying logarithmic expressions. It allows you to move the exponent in the argument of a logarithm to the front as a multiplier. This rule is expressed mathematically as \( \log_b(a^c) = c \cdot \log_b(a) \). Using this rule can greatly simplify complex expressions, turning them into more manageable forms.
For instance, consider \( \log_2(y^x) \). Here, \( y^x \) represents \( a^c \), where \( y \) is the base and \( x \) is the exponent or power. According to the power rule, this expression can be expanded to \( x \cdot \log_2(y) \).
This conversion allows for standard operations on logarithms, making calculations simpler and more efficient.

Properties of Logarithms

Logarithms have several core properties that aid in mathematical manipulation. These properties help in simplifying logarithmic expressions, solving equations, and performing calculations.
Here are the main properties of logarithms:

• Product Property: The logarithm of a product is the sum of the logarithms. Mathematically, \( \log_b(mn) = \log_b(m) + \log_b(n) \).
• Quotient Property: The logarithm of a quotient is the difference of the logarithms. In formula terms, \( \log_b(\frac{m}{n}) = \log_b(m) - \log_b(n) \).
• Power Rule: As mentioned, it states \( \log_b(m^n) = n \cdot \log_b(m) \).

Each property serves to transform logarithmic expressions into a preferred form for the problem at hand. Understanding these properties is essential for anyone studying algebra and calculus.

Exponential Functions

An exponential function is one where the variable appears in the exponent. These functions can take the form \( f(x) = b^x \), where \( b \) is a constant called the base, and \( x \) is the exponent. Exponential functions are foundational in mathematics and they appear in various scientific disciplines because they describe growth and decay processes.
Exponential functions are characterized by:

• A constant base raised to a variable exponent.
• A rapidly changing rate, which can model population growth, radioactive decay, and more.

Understanding their interaction with logarithms is crucial because logarithms are essentially the inverse operations of exponentiation. This relationship allows us to solve exponential equations by converting them into simpler logarithmic ones.