Problem 22 - Algebra 2 and Trigonometry

Question

In \(11-22,\) solve each equation for \(y\) in terms of \(x\) $$ x=\log _{0.1} y $$

Step-by-Step Solution

Answer

The solution is \( y = 0.1^x \).

Step 1: Understand the Logarithm

The given equation is \( x = \log_{0.1} y \). This equation represents a logarithm where \( x \) is the power to which \( 0.1 \) must be raised to get \( y \).

Step 2: Convert Logarithmic Equation to Exponential Form

The logarithmic equation \( x = \log_{0.1} y \) can be rewritten in exponential form as \( y = 0.1^x \). This transformation uses the property that \( \log_b a = c \) is equivalent to \( a = b^c \).

Step 3: Equation Simplification

Since \( y = 0.1^x \) is the expression for \( y \) in terms of \( x \), we conclude that this is the simplest form of the solution.

Key Concepts

Logarithmic Equations

Logarithmic equations involve the concept of logarithms, which are the inverse operations of exponentiation. In the case of the given equation \( x = \log_{0.1} y \), \( x \) represents the power to which the base, \( 0.1 \), must be raised to obtain \( y \). This relationship is a crucial aspect of understanding logarithms as a whole. Key characteristics of logarithms include:

• The base indicates the number being raised to a power.
• The logarithm itself represents the exponent or power.
• When the base is less than one, as in this case with \( 0.1 \), the values of \( y \) will become smaller as \( x \) increases.

Understanding how to interpret and manipulate these characteristics is vital when solving logarithmic equations for unknowns.

Exponential Form

Transitioning from a logarithmic equation to its exponential form helps simplify and solve equations. To transform a logarithmic equation like \( x = \log_{0.1} y \) into an exponential form, we use the fundamental principle that a logarithm base \( b \) raised to the power \( x \) equals \( y \), written as \( y = b^x \). Here, the base is \( 0.1 \), resulting in the equation \( y = 0.1^x \).This exponential form makes it clearer by expressing \( y \) directly as a function of \( x \). This format is both a useful manipulation tool and helps us in visualizing the behavior of the function. Traits of the exponential form include:

• Converting complex logarithmic expressions into simple power equations.
• Facilitating a straightforward understanding of changes in \( y \) relative to \( x \).

These characteristics illustrate why shifting to an exponential form is a powerful step in solving equations.

Algebraic Transformations

Algebraic transformations are techniques used to rearrange and solve equations. Transforming a logarithmic equation into an exponential form is one such transformation, enabling easier solving. In our example, converting \( x = \log_{0.1} y \) into \( y = 0.1^x \) simplifies the equation, showcasing how core algebraic manipulation can effectively solve for \( y \) in terms of \( x \).Imagine algebraic transformations as puzzle pieces, helping make complex equations more manageable. They often include:

• Rewriting equations in friendlier formats.
• Isolating variables to solve for unknowns.
• Simplifying expressions to reveal underlying relationships.

By mastering algebraic transformations, you enhance your ability to work with, understand, and solve varied mathematical equations, beyond just logarithmic functions.