Problem 45 - Elementary and Intermediate Algebra

Question

Solve each equation. See Example \(6 .\) $$ \log (7-x)=2 $$

Step-by-Step Solution

Answer

The solution to the equation is \(x = -93\).

Step 1: Understand the Logarithm

In this equation, we have \(\log(7-x) = 2\). This means that the logarithm base 10 of \(7-x\) is equal to 2. We want to isolate \(x\) by removing the logarithm.

Step 2: Convert to Exponential Form

Recall that \(\log_{10}(a) = b\) is equivalent to \(a = 10^b\). Therefore, \(\log(7-x) = 2\) translates to the exponential equation \(7-x = 10^2\).

Step 3: Simplify the Exponential Equation

Calculate \(10^2\), which equals 100. Now we have the equation \(7-x = 100\).

Step 4: Isolate \(x\)

To solve for \(x\), subtract 7 from both sides of the equation: \(-x = 100 - 7\). Simplifying, we have \(-x = 93\).

Step 5: Solve for Positive \(x\)

Multiply both sides by -1 to solve for \(x\): \(x = -93\).

Key Concepts

Logarithmic Functions

Logarithmic functions are essential in understanding the relationship between exponential and logarithmic expressions. They are the inverse of exponential functions. In simple terms, the logarithm of a number is the exponent by which the base must be raised to produce that number. In the equation \(\log(7-x) = 2\), the base of the logarithm is 10, which is typically assumed when not explicitly stated. Therefore, what this tells us is that 10 raised to the power of 2 equals \(7-x\). This inverse relationship allows us to manipulate logarithmic expressions and convert them to exponential form, making equations easier to solve.

Exponential Equations

Converting a logarithmic equation into an exponential form is a crucial step in solving for the unknown. It requires us to understand that in the equation \(\log(7-x) = 2\), we can rewrite it as an exponential equation: \(7-x = 10^2\). Here:

• The base of the exponent is 10 (from the original log base).
• The exponent in the exponential form is 2 (the value the log equals).
• The result of the expression \(10^2\) results in the value that \(7-x\) must equal.

This step enables us to break down and solve the equation more straightforwardly, transitioning from a logarithmic form to an arithmetic problem that can be simplified.

Algebraic Manipulation

Once the equation is in the form \(7-x = 100\), algebraic manipulation becomes straightforward. The goal is to isolate \(x\) on one side of the equation. Start by moving the constant from the side with \(x\) by subtracting 7 from both sides:

• \(-x = 100 - 7\)
• Simplify to get \(-x = 93\).

Finally, to solve for \(x\), multiply the entire equation by -1 to find \(x = -93\). This manipulation process follows typical algebra rules and is essential for isolating variables in various forms of equations. By applying these steps carefully, we can find the value that satisfies the original equation efficiently.