Problem 42 - Algebra and Trigonometry

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 Logarithmic Equation

The given equation is \( \log _{3}(2-x) = 3 \). This means that \( 2-x \) is equal to \( 3^3 \). We need to solve for \( x \).

Step 2: Convert the Logarithmic Equation to Exponential Form

The equation \( \log _{3}(2-x) = 3 \) can be rewritten in exponential form as \( 2-x = 3^3 \).

Step 3: Simplify the Exponential Expression

Calculate \( 3^3 \), which is \( 27 \). So, the equation becomes \( 2-x = 27 \).

Step 4: Solve the Linear Equation for x

Rearrange the equation \( 2-x = 27 \) to find \( x \). Subtract 2 from both sides to get \( -x = 27 - 2 \). This simplifies to \( -x = 25 \).

Step 5: Isolate x

Multiply both sides of \( -x = 25 \) by \(-1\) to solve for \( x \). This gives \( x = -25 \).

Key Concepts

Exponential Form

When dealing with logarithmic equations, one of the first steps is to convert the logarithmic expression into its exponential form. The logarithmic equation given is \( \log_{3}(2-x) = 3 \). This can be read as "the power you have to raise 3 to, in order to get \(2-x\), is 3."

To switch from logarithmic to exponential form, remember that \( \log_{b}(a) = c \) is equivalent to saying \( b^c = a \). For our problem, \( b \) is 3 and \( c \) is 3, thus:

• Convert \( \log_{3}(2-x) = 3 \) to \( 3^3 = 2-x \).

Recognizing and using this conversion is fundamental to solving a problem involving a logarithm. It often transforms a complex logarithmic equation into a simpler exponential equation.

Once in exponential form, the equation is easier to manipulate and solve, as it becomes a simple matter of evaluating powers and rearranging terms.

Linear Equation

After converting the logarithmic equation into the exponential form, the resulting equation is \( 2-x = 27 \). This is a straightforward linear equation, which is an equation involving no exponents and resembles the standard form \( ax + b = c \).

For this problem, our task is to isolate \( x \). A helpful strategy for solving linear equations involves rearranging terms:

• Subtract constants from both sides of the equation first. Here, subtract 2 from both sides.
• Convert \( 2 - x = 27 \) into \( -x = 27 - 2 \).

Linear equations like these often require simple arithmetic operations. Carefully following each step ensures that we maintain the equation's balance, an essential aspect of solving these types of equations.

Solving Equations

The key part of solving equations is to isolate the variable. After simplifying the exponential form and rearranging the linear equation to \( -x = 25 \), the final step is to solve for \( x \).

To isolate \( x \), we need to get rid of the negative sign. This is done by multiplying or dividing each side of the equation by the factor of the variable:

• Multiply each side by \(-1\): \(-x \times (-1) = 25 \times (-1)\).
• This operation results in \( x = -25 \).

Multiplication or division of both sides by the same non-zero number is a common solution step. It helps make equations easier to solve and ensures we achieve the simplest form of the solution.