Problem 42 - Precalculus Mathematics for Calculus

Question

Solve the logarithmic equation for \(x .\) $$\log _{3}(2-x)=3$$

Step-by-Step Solution

Answer

The solution is \( x = -25 \).

Step 1: Understand the Problem

We need to find the value of \( x \) that satisfies the equation \( \log_{3}(2-x) = 3 \). This means that the logarithm base 3 of \( 2-x \) is 3.

Step 2: Convert Logarithmic Equation to Exponential Form

Recall that the equation \( \log_{b}(a) = c \) can be reformulated in exponential form as \( a = b^c \). Therefore, \( 2-x = 3^3 \).

Step 3: Simplify the Exponential Expression

Calculate \( 3^3 \), which is \( 27 \). Therefore, the equation simplifies to \( 2-x = 27 \).

Step 4: Solve for x

Subtract 2 from both sides of the equation to isolate \( -x \): \[ -x = 27 - 2 \] Simplifying, we get \( -x = 25 \).

Step 5: Find the Value of x

To solve for \( x \), multiply both sides by -1: \[ x = -25 \].

Key Concepts

Logarithmic Functions

Logarithmic functions are the inverses of exponential functions. They come in handy when we want to solve equations involving powers or exponential growth. A logarithmic function with a base of 'b' is expressed as \( \log_b(x) \), where 'b' is a constant greater than 1, and \( x \) is a positive real number.
This function answers the question: "What exponent do I need on 'b' to get 'x'?" For example, if we have \( \log_3(9) \), we are asking, "3 raised to what power gives us 9?" We find that it is 2, because \( 3^2 = 9 \).
In general, logarithmic functions are used in many real-world scenarios, such as calculating the intensity of earthquakes (Richter scale), pH in chemistry, and even in finance for compound interest scenarios.

• Logarithms simplify multiplication into addition.
• They turn powers into multiplication.
• Logarithms help in solving exponential equations.

Exponential Equations

Exponential equations are equations where variables appear as exponents. These equations are often challenging to solve because they can involve very large or small numbers due to the nature of exponentiation.
For example, in our original problem, we transformed the logarithmic equation \( \log_{3}(2-x) = 3 \) into an exponential equation, \( 2-x = 3^3 \). This transition is crucial, as it allows us to deal with equal expressions without logarithms.
Exponential equations can typically be solved in several ways, including converting them into a logarithmic form, or by using properties of exponents:

• When bases are the same on both sides, set the exponents equal to each other.
• Utilize logarithms to "bring down" the exponent.

Exponential equations show up in many daily circumstances, such as computing compound interest, planning population growth, and even analyzing the process of radioactive decay.

Solving Equations

Solving equations is at the heart of mathematics. It involves finding the values of variables that make the equation true. The approach taken depends on the type of equation, such as linear, exponential, or logarithmic.
In our example, solving the equation involved several steps: converting a logarithmic equation to an exponential form and then isolating the variable \( x \) to find its value. This method requires understanding both the divisional nature of logarithms and the multiplicative nature of exponents.
General steps to solve equations include:

• Identify the type of equation.
• Use appropriate conversion methods (e.g., logarithmic to exponential).
• Rearrange to isolate the variable.
• Simplify the equation to find the solution.

These steps are helpful not only in mathematics but in everyday problem-solving, helping to break down complex problems into manageable parts.

Base of Logarithm

The base of a logarithm is a crucial component of a logarithmic function. It tells us the number we repeatedly multiply to get a certain value. In the expression \( \log_b(a) \), 'b' is the base.
Choosing a base depends on the context:

• Base 10 (common logarithms) is prevalent in scientific calculations and pH measurements.
• Base \( e \) (natural logarithms) is used in continuous growth models, like radioactive decay or population growth.
• Specific bases, like base 2, are often used in computer science because of binary operations.

Understanding the base of a logarithm is essential because it determines how the function behaves and influences how equations are solved. In our problem, the base 3 was used, indicating the number used exponentially to reach another number, namely \( 2-x \). This concept is vital for interpreting and using logarithmic equations effectively.