Problem 7 - Algebra and Trigonometry

Question

Find the solution of the exponential equation, rounded to four decimal places. \(2^{1-x}=3\)

Step-by-Step Solution

Answer

The solution is approximately \(x = -0.5850\).

Step 1: Understand the Problem

We are given the exponential equation \(2^{1-x} = 3\) and we need to find the value of \(x\) such that the equation holds true. This requires us to solve for \(x\) using mathematical operations.

Step 2: Take the Logarithm of Both Sides

To solve for \(x\), start by taking the logarithm of both sides of the equation \(2^{1-x} = 3\). For simplicity, we will use the natural logarithm (\(\ln\)):\[\ln(2^{1-x}) = \ln(3)\] This allows us to use the power rule of logarithms, \(\ln(a^b) = b \cdot \ln(a)\), to bring the exponent down.

Step 3: Apply the Power Rule of Logarithms

Using the power rule, we simplify:\[(1-x) \cdot \ln(2) = \ln(3)\]Now, we can isolate \(x\) by further manipulation.

Step 4: Solve for x

To isolate \(x\), divide both sides by \(\ln(2)\):\[1-x = \frac{\ln(3)}{\ln(2)}\]Then, solve for \(x\) by subtracting the fraction from 1:\[x = 1 - \frac{\ln(3)}{\ln(2)}\]

Step 5: Compute the Solution

Now, compute the expression using a calculator:\[\ln(3) \approx 1.098612 \quad \text{and} \quad \ln(2) \approx 0.693147\]So:\[\frac{\ln(3)}{\ln(2)} \approx \frac{1.098612}{0.693147} \approx 1.58496\]Finally:\[x = 1 - 1.58496 \approx -0.58496\]Rounding to four decimal places, the solution is \(x \approx -0.5850\).

Key Concepts

Logarithms

Logarithms are an essential tool in mathematics, allowing us to solve equations involving exponents by bringing down the powers. When you encounter an exponential equation like \(2^{1-x} = 3\), your first step is often to apply logarithms to both sides. This technique transforms the equation into a form where the exponent can be isolated.

• The logarithm of a number is the exponent to which the base of the logarithm must be raised to produce that number.
• You can use logarithms with any base, but common bases include base 10 (common logarithm) and base \(e\) (natural logarithm).

In the given problem, using the natural logarithm simplifies the calculation and allows you to proceed with solving the equation.

Natural Logarithm

The natural logarithm, denoted as \(\ln\), is particularly popular in solving exponential equations. It's based on the mathematical constant \(e\), which is approximately 2.71828. In contexts involving exponential growth and decay, the natural logarithm is frequently used. Applying the natural logarithm to both sides of the equation \(2^{1-x} = 3\) leads to:\[\ln(2^{1-x}) = \ln(3)\]

• Using \(\ln\) helps simplify equations because of its powerful properties, allowing you to bring down exponents.
• Many scientific and engineering applications prefer \(\ln\) since it naturally arises in continuous growth models.

Solving this part of the equation sets the stage for using the power rule, which is another significant logarithmic property.

Power Rule of Logarithms

One of the critical properties of logarithms is the power rule. The power rule states that for any real number \(b\) and positive number \(a\), \(\ln(a^b) = b \cdot \ln(a)\). This property is enormously helpful as it allows you to deal with exponents in equations easily. In our problem, we applied this rule to the expression \(\ln(2^{1-x})\), which becomes:\[(1-x) \cdot \ln(2) = \ln(3)\]

• The power rule transforms complex exponential expressions into simpler linear ones.
• This transformation enables solutions through straightforward algebraic manipulation.

Once simplified, you can isolate the variable \(x\), ultimately finding the value that solves the original equation.