Problem 37 - A Graphical Approach to College Algebra

Question

Solve each logarithmic equation. Express all solutions in exact form. Support your solutions by using a calculator. $$\log (2-x)=0.5$$

Step-by-Step Solution

Answer

The solution is \( x = 2 - \sqrt{10} \).

Step 1: Understand the Logarithmic Equation

We are given the equation \( \log (2-x) = 0.5 \). Here, the base of the logarithm is assumed to be 10. We will rewrite this equation in its exponential form to solve for \(x\).

Step 2: Convert to Exponential Form

Recall that an equation of the form \( \log_b(y) = x \) can be rewritten in exponential form as \( y = b^x \). For our equation \( \log(2-x) = 0.5 \), this translates to \( 2-x = 10^{0.5} \).

Step 3: Calculate the Exponent Value

Calculate \( 10^{0.5} \). Since 0.5 is the square root of \( 10 \), \( 10^{0.5} = \sqrt{10} \). Therefore, we have \( 2-x = \sqrt{10} \).

Step 4: Solve for \(x\)

We now solve for \(x\) in the equation \( 2-x = \sqrt{10} \). Rearrange the equation to solve for \(x\):\[ x = 2 - \sqrt{10} \].

Step 5: Verify the Solution Using a Calculator

Use a calculator to approximate \( 2 - \sqrt{10} \). Calculate \( \sqrt{10} \approx 3.1623 \), then \( 2 - 3.1623 \approx -1.1623 \). Verify that substituting \( x = 2 - \sqrt{10} \) into the original logarithmic expression confirms the solution is correct.

Key Concepts

Exponential Form

The concept of converting logarithmic equations to exponential form is essential when solving logarithmic problems. A logarithm is essentially the inverse of an exponentiation. It answers the question, "To what exponent must the base be raised, to yield the number?" For example, in a logarithmic equation of the form \( \log_b(y) = x \), \(b\) is the base, \(y\) is the result of raising \(b\) to the power of \(x\). This can be reformulated as \( y = b^x \) in exponential terms.
In our given problem, \( \log(2-x)=0.5 \) is transformed into \( 2-x=10^{0.5} \). Here, since no specific base is indicated for the logarithm, we assume a common base of 10. Thus, the exponential form becomes \( 2-x = 10^{0.5} \), simplifying the equation and making it solvable for \(x\).
Converting to exponential form is a powerful step because it simplifies equations, allowing you to solve for the unknown variable effectively.

Solving Equations

Once an equation is in its exponential form, the next step is solving for the unknown variable. For the equation \( 2-x = \sqrt{10} \), you need to isolate \(x\). This is done by simply rearranging the equation.
- Start with \( 2-x = \sqrt{10} \).
- Rearrange to solve for \(x\): \(x = 2 - \sqrt{10}\).
This straightforward rearrangement isolates the variable of interest, \(x\), giving us a precise form of the solution. Using numerical approximations is helpful for understanding the magnitude of the solution. Here, \( \sqrt{10} \approx 3.1623\), resulting in \(x \approx 2 - 3.1623 = -1.1623\).
This step not only finds the solution but confirms it by allowing easy substitution back into the original equation for verification.

Properties of Logarithms

Logarithms possess unique properties that make them extremely useful in solving equations, especially those involving exponential relationships. Some key properties include:

• The product rule: \( \log_b(MN) = \log_b(M) + \log_b(N) \)
• The quotient rule: \( \log_b\left(\frac{M}{N}\right) = \log_b(M) - \log_b(N) \)
• The power rule: \( \log_b(M^k) = k \cdot \log_b(M) \)

These properties allow complex logarithmic expressions to be simplified. However, in our specific problem, these properties help in setting the ground for understanding the shift from logarithmic to exponential form if needed. Knowing these can simplify many steps in working through logarithmic equations and translating them into exponential equations.
Each property aids in dismantling complicated logarithmic expressions, merging multiple logarithms into single expressions, or vice versa, making the process of solving log-based equations more tractable.