Problem 43 - Precalculus Mathematics for Calculus

Question

Solve the logarithmic equation for \(x .\) $$4-\log (3-x)=3$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = -7\).

Step 1: Isolate the Logarithmic Term

To solve the equation \(4 - \log (3-x) = 3\), we first need to isolate the logarithmic term. We do this by subtracting 4 from both sides of the equation:\[4 - \log (3-x) - 4 = 3 - 4\]Simplifying gives:\[-\log (3-x) = -1\]

Step 2: Remove the Negative Sign

Next, removing the negative sign from the logarithm is necessary. This can be done by multiplying both sides of the equation by -1:\[\log (3-x) = 1\]

Step 3: Convert Logarithm to Exponential Form

To solve for \(x\), convert the logarithmic equation into exponential form. Remember that if \(\log_b(a) = c\), then \(a = b^c\). As the equation is in base 10, this becomes:\[3-x = 10^1\]which simplifies to:\[3-x = 10\]

Step 4: Solve for x

Finally, solve \(3-x = 10\) for \(x\). We do this by moving \(x\) to the other side and then solving:\[3 - 10 = x\]\[x = -7\]

Key Concepts

Exponential Form

When working with logarithmic equations, a crucial step in solving them is understanding how to convert them into their exponential form. The exponential form is a different way of expressing the same relationship that a logarithm shows, but in terms of exponentiation. A basic rule to remember is:

• If you have a logarithmic equation \(\log_b(a) = c\),
• It can be rewritten in its exponential form as \(a = b^c\).

Converting to this form can greatly simplify an equation and make it easier to solve.
In the exercise, \(\log(3-x) = 1\), the base is 10 (as is usual in common logarithms). This means we convert it to \(3-x = 10^1\), which simply calculates to 10. This step is often a turning point in making the equation straightforward to solve. Always check the base before converting and proceed step-by-step to avoid errors.

Logarithmic Functions

Logarithmic functions are an essential part of mathematics, especially when dealing with multiplicative processes and exponential growth scenarios. A logarithm answers "what power must we raise a number (the base) to obtain another number?" Here's what to keep in mind:

• The expression \(\log_b(a) = c\) means \(b^c = a\).
• Logarithms help in undoing exponentials and are used in various applications like calculating time for growth/decay processes.

For our example, \(\log(3-x) = 1\) suggests that if we raise 10 to the power of 1, we get \(3-x\). This transformation of the equation reveals the hidden exponential relationship and helps bring out the value to be solved.
Recognizing the type of logarithm and accurately performing operations with these functions are key skills in dealing with these kinds of problems effectively.

Isolating Terms

Isolating terms is a common strategy used in solving equations, whether they involve logarithms or not. The main idea is to rearrange the equation so that the variable of interest is alone on one side of the equation. Here’s a breakdown of the process:

• Identify and separate the term that involves the variable.
• Perform operations to move other terms to the opposite side, usually addition or subtraction.
• Finally, take necessary actions to solve the remaining simplified equation.

In our equation, \(4 - \log(3-x) = 3\), the goal is to isolate the term \(-\log(3-x)\). Start by subtracting 4 from both sides, resulting in \(-\log(3-x) = -1\). Then, remove the negative sign by multiplying by -1. This yields \(\log(3-x) = 1\), making it ready for transformation into exponential form.
Always aim for logical and straightforward steps when isolating terms. It simplifies the equation and helps avoid any unnecessary complexities.