Problem 17 - Precalculus

Question

Write each logarithmic equation in its equivalent exponential form. $$z=\log _{x} y$$

Step-by-Step Solution

Answer

The equivalent exponential form is \( x^z = y \).

Step 1: Understand the Logarithmic Form

The given equation is in the logarithmic form, which is \( z = \log_x y \). In this form, \( x \) is the base, \( y \) is the result, and \( z \) is the exponent that the base must be raised to, in order to yield the result.

Step 2: Convert to Exponential Form

To convert the logarithmic equation \( z = \log_x y \) to an exponential equation, recall the relationship between logarithms and exponents: if \( z = \log_x y \), then the equivalent exponential form is \( x^z = y \).

Step 3: Write the Final Exponential Equation

Based on the conversion from the previous step, the exponential form of the given logarithmic equation is \( x^z = y \). This equation represents the same relationship as the logarithmic form, but expressed as an exponentiation problem.

Key Concepts

Exponential Form

When working with logarithmic equations, converting them into exponential form is a fundamental skill. Let's begin by understanding what exponential form means.
The exponential form of an equation involves expressing a number as a power of another number. It generally looks like this:

• Base: The number that is repeatedly multiplied;
• Exponent: The power to which the base is raised;
• Result: The outcome of raising the base to the given exponent.

Understanding the structure of exponential expressions helps us solve various mathematical problems. In the case of the equation we've been given from the exercise, converting the logarithmic expression helps unveil the base and the exponent's roles.
In exponential form, a typical expression would be arranged as \(x^z = y\), where \(x\) is the base, \(z\) is the exponent, and \(y\) is the result. Learning to identify and adjust these parts is crucial for mastering both logarithmic and exponential equations.

Base and Exponent Relationship

The base and exponent relationship is pivotal when dealing with logarithms and their transformations into exponentials.
Think of the base as the foundation of your equation; it provides the number that you'll raise to the power specified by the exponent. The exponent then dictates how many times the base is multiplied by itself.For instance, in the expression \(x^z = y\), understanding this relationship is key:

• The base \(x\) is your starting number;
• The exponent \(z\) directs how many times to multiply the base \(x\) by itself;
• The result \(y\) represents the outcome of this process.

Grasping this relationship simplifies both solving problems and understanding functions' behavior, such as exponential growth or decay. This foundation of the base and exponent is critical for converting between different mathematical forms effectively.

Logarithmic and Exponential Conversion

Logarithmic and exponential conversions are interlinked processes that help solve complex mathematical equations easily.
The relationship can be summarized: a logarithmic function \(z = \log_x y\) is converted into an exponential function as \(x^z = y\). The conversion follows a simple thought process where:

• Identify your base \(x\) in the logarithmic part;
• Recognize the exponent \(z\), which will dictate the power;
• Understand that the result, \(y\), is what the base raised to the exponent equals.

Converting between these forms allows problem-solving in different contexts, offering an alternate perspective to view the same relationship.
This form of algebraic translation serves as a fundamental tool in mathematics, simplifying problems by allowing them to be approached from a different angle. Mastering these conversions is crucial for tackling more advanced topics in calculus, engineering, and physics.