Problem 72 - Algebra 2

Question

Write an equivalent logarithmic equation. $$ b^{y}=x $$

Step-by-Step Solution

Answer

The equivalent logarithmic equation is \( y = \log_b(x) \).

Step 1: Identify the Components

To convert the exponential equation to a logarithmic equation, first identify the components: the base, the exponent, and the result. In the equation \( b^y = x \), \( b \) is the base, \( y \) is the exponent, and \( x \) is the result.

Step 2: Apply the Logarithmic Conversion

The general form to convert an exponential equation \( b^y = x \) into a logarithmic equation is \( y = \log_b(x) \). Here, the exponent \( y \) becomes isolated, indicating it equals the logarithm of the result \( x \) with respect to base \( b \).

Step 3: Write the Equivalent Logarithmic Equation

Using the form from the previous step, rewrite the given equation as a logarithmic equation: \( y = \log_b(x) \).

Key Concepts

Exponential Equations

Exponential equations are mathematical expressions where a variable appears in the exponent. These equations take the form \( b^y = x \), where \( b \) is the base, \( y \) is the exponent, and \( x \) is the resultant value. Exponential equations are often encountered in real-world applications such as compound interest calculations, population growth models, and scientific phenomena involving exponential growth or decay. For instance, if you invest a certain amount of money at a fixed interest rate, the total amount of money grows exponentially over time.

When solving exponential equations, one key objective is often to find the value of the exponent \( y \), which explains how many times the base \( b \) is multiplied by itself to yield the result \( x \). As we proceed to explore these equations, understanding how to isolate the exponent with techniques such as logarithmic conversion will prove valuable.

Logarithmic Conversion

Logarithmic conversion is an essential technique used to transform an exponential equation into a logarithmic form, making it easier to solve for the exponent. In the context of exponential equations, where we have \( b^y = x \), logarithmic conversion changes it into \( y = \log_b(x) \). This conversion tells us that the exponent \( y \) is the power to which the base \( b \) must be raised to produce the result \( x \). This transformation is founded on the definition of logarithms.

Some key points to remember when performing logarithmic conversion are:

• The base \( b \) of the exponent becomes the base of the logarithm.
• The exponent \( y \) is what you solve for, and it becomes the left-hand side of the equation.
• The result \( x \) goes inside the logarithm with base \( b \).

This technique simplifies the process of solving exponential equations, especially when the exact value of \( y \) is unknown or challenging to evaluate directly.

Base and Exponent Identification

Before performing operations like logarithmic conversion, it's crucial to clearly identify the base and exponent in an exponential equation. In the equation \( b^y = x \), \( b \) is the base and shows the number continually multiplied, while \( y \) represents how many times the base is multiplied by itself. The quantity \( x \) results from these repeated multiplications.

Properly identifying these components makes it easier to perform logarithmic conversions and comprehend the relationship between the base, the result, and the exponent. Here are a few tips for identifying these elements:

• The base \( b \) is usually the number or expression that is raised to a power.
• The exponent \( y \) is the power to which the base is raised, determining the multiplication process.
• The result \( x \) is what you arrive at after applying the power to the base.

Being adept at base and exponent identification aids in demystifying complex exponential relations and eases the transition to logarithmic expressions.