Problem 28 - Precalculus with Calculus Previews

Question

Solve the given logarithmic equation. $$ \log _{5}|1-x|=1 $$

Step-by-Step Solution

Answer

The solutions are \( x = -4 \) and \( x = 6 \).

Step 1: Understand the Logarithmic Equation

The given equation is \( \log_{5}|1-x| = 1 \). Here, the equation represents a logarithm with base 5 and the magnitude of \( 1-x \) as its argument. We are tasked with finding the values of \( x \) that satisfy this equation.

Step 2: Convert Logarithmic Equation to Exponential Form

To solve the logarithmic equation, convert it to its exponential form. The equation \( \log_{5}|1-x| = 1 \) can be rewritten as \( |1-x| = 5^1 \). This simplifies to \( |1-x| = 5 \).

Step 3: Solve the Absolute Value Equation

The equation \( |1-x| = 5 \) means that the absolute value of \( 1-x \) is 5, leading to two possible cases:1. \( 1-x = 5 \)2. \( 1-x = -5 \)

Step 4: Solve Each Case Separately

For the first case, solve \( 1-x = 5 \): - Subtract 1 from both sides: \(-x = 4\)- Multiply both sides by -1 to solve for \( x \): \( x = -4 \)For the second case, solve \( 1-x = -5 \):- Subtract 1 from both sides: \(-x = -6\)- Multiply both sides by -1 to solve for \( x \): \( x = 6 \)

Step 5: Verify Solutions in the Context of the Original Equation

The possible solutions are \( x = -4 \) and \( x = 6 \). Verify these by substituting back into \( |1-x| = 5 \):- For \( x = -4 \), \( |1 - (-4)| = |1 + 4| = 5 \)- For \( x = 6 \), \( |1 - 6| = |-5| = 5 \)Both solutions satisfy \( |1-x| = 5 \), and therefore satisfy the original equation.

Key Concepts

Exponential Form

When dealing with logarithmic equations, understanding the connection to exponential form is crucial. The logarithmic equation you are working with is \( \log_{5}|1-x| = 1 \), which tells you how many times the base, 5, must be multiplied by itself to result in the number inside the log function. Converting it into exponential form follows a straightforward rule: If \( \log_{b}(A) = C \), then \( b^C = A \).

In our example, the base of the logarithm is 5, and the exponential form of the equation becomes \( |1-x| = 5^1 \). Since \( 5^1 = 5 \), we have the expression \( |1-x| = 5 \). The exponential form aids in simplifying and solving the equation by getting rid of the logarithmic function and providing a clear numerical target (5 in this case).

Absolute Value Equation

The equation \( |1-x| = 5 \) represents what we call an "absolute value equation." Absolute value considers only the magnitude of a number, without regard to its sign. Thus, \( |1-x| = 5 \) implies that the difference between 1 and \( x \) could be either +5 or -5.

To solve an absolute value equation, split it into two separate cases based on the definition of absolute value:

• Case 1: \( 1-x = 5 \)
• Case 2: \( 1-x = -5 \)

By solving each scenario individually, we ensure all possible solutions are considered. This step is essential, as it accounts for both the positive and negative differences that satisfy the original equation.

Base of a Logarithm

The base of a logarithm is a critical part of understanding logarithmic equations. In the equation \( \log_{5}|1-x| = 1 \), the number 5 is the base of the logarithm. The base 5 not only indicates how many times the base needs to be multiplied to get its argument, but it also directly affects the calculation in the exponential form.

Different bases can completely change the equation and solution. If the base were different, such as 2 or 10, the exponential form \( |1-x| \) would also change to \( 2^1 \) or \( 10^1 \) respectively. Therefore, recognizing what the base signifies is a foundational skill in handling logarithmic expressions.

Solution Verification

Solution verification is the final step to ensure that the solutions found actually satisfy the original equation. After calculating the values \( x = -4 \) and \( x = 6 \) from the absolute value equations, it’s vital to plug these back into the context of the original logarithmic equation to verify them.

Check each solution:

• For \( x = -4 \), calculate: \( |1 - (-4)| = |1 + 4| = 5 \). The calculation confirms \( 5 = 5 \), verifying the solution.
• For \( x = 6 \), calculate: \( |1 - 6| = |-5| = 5 \). Again, this confirms \( 5 = 5 \), verifying this solution too.

This process of verification ensures no mistakes were made during calculations and that all provided solutions are indeed correct in satisfying the initial logarithmic equation.