Problem 34 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 19-34: Solve the equation. $$ e^{x \ln 2}=0.25 $$

Step-by-Step Solution

Answer

The solution to the equation is \( x = -2 \).

Step 1: Recognize the Exponential Equation

The given equation is \( e^{x \ln 2} = 0.25 \). This is an exponential equation where the exponent is expressed as a product of \( x \) and \( \ln 2 \).

Step 2: Apply the Natural Logarithm

To solve for \( x \), take the natural logarithm of both sides of the equation. Applying \( \ln \) gives us: \[ \ln(e^{x \ln 2}) = \ln(0.25) \] Using the property of logarithms, \( \ln(a^b) = b \cdot \ln(a) \), we simplify the left side to obtain:\[ x \ln 2 = \ln(0.25) \]

Step 3: Solve for x

Now that we have the equation \( x \ln 2 = \ln(0.25) \), solve for \( x \) by dividing both sides by \( \ln 2 \): \[ x = \frac{\ln(0.25)}{\ln 2} \] Remember that \( \ln(0.25) = \ln\left(\frac{1}{4}\right) = -\ln(4) \), and since \( 4 = 2^2 \), we have \( \ln(4) = 2 \ln(2) \). Thus, substitute to find:\[ x = \frac{-2 \ln 2}{\ln 2} = -2 \]

Step 4: Verify the Solution

Now we verify by substituting \( x = -2 \) back into the original equation: \[ e^{-2 \ln 2} = (e^{\ln 2})^{-2} = 2^{-2} = \frac{1}{4} = 0.25 \] This confirms that the solution \( x = -2 \) is correct.

Key Concepts

Natural Logarithm

The natural logarithm, denoted as \( \ln \), is a logarithm to the base \( e \), where \( e \) is an irrational constant approximately equal to 2.71828. The natural logarithm is a fundamental concept in mathematics, especially in calculus and logarithmic computations. When you apply \( \ln \) to an expression, you are essentially asking, "To what power must I raise \( e \) to get this number?"

For example, in the context of solving exponential equations like \( e^{x \ln 2} = 0.25 \), applying \( \ln \) allows us to simplify and linearize the equation. By taking the natural logarithm of both sides:

• We can use properties of logarithms to bring down exponents, turning them into coefficients.
• This technique simplifies the manipulation and solving of exponential equations drastically.

Properties of Logarithms

Logarithms, including the natural logarithm, have properties that make them useful in solving various mathematical problems, especially exponential equations. The key properties include:

• Power Rule: \( \ln(a^b) = b \cdot \ln(a) \). This allows exponents to be converted into coefficients, simplifying complex expressions.
• Product Rule: \( \ln(ab) = \ln(a) + \ln(b) \). This is useful when multiplying numbers within a logarithm.
• Quotient Rule: \( \ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b) \). Applicable when dividing numbers inside a log.

By utilizing these properties, you can efficiently manipulate and solve equations. For instance, in our problem, we used the power rule to convert the expression \( e^{x \ln 2} = e^{\ln(0.25)} \) into a simpler form for \( x \ln 2 = \ln(0.25) \).

These properties transform complex equations into manageable algebraic expressions.

Solving for Variables

Solving for variables in equations, especially exponential ones, often involves isolating the variable of interest. In problems like \( e^{x \ln 2} = 0.25 \), once logarithms have been applied and simplified, the next step involves isolating \( x \).

This process typically includes:

• Using algebraic manipulation such as division, multiplication, or additional inverse operations.
• Applying log properties to further simplify equations and isolate the variable.

To isolate \( x \), after simplifying \( x \ln 2 = \ln(0.25) \), you divide each side by \( \ln 2 \), yielding \( x = \frac{\ln(0.25)}{\ln(2)} \).

By breaking down each step and applying mathematical properties systematically, the process becomes much simpler. In the end, this leads to a numerical determination of \( x \) that can be checked by substitution back into the original equation. This ensures accuracy and reinforces understanding of both the technique and the result. By mastering these techniques, solving exponential equations becomes a structured and predictable process.