Problem 21 - Beginning and Intermediate Algebra

Question

Solve the exponential equation algebraically. Then check using a graphing calculator. $$3^{x}=2^{x-1}$$

Step-by-Step Solution

Answer

In short, to solve the exponential equation \(3^{x}=2^{x-1}\), we first took the natural logarithm of both sides and then used the properties of logarithms to rewrite the equation as \(x\ln(3) = (x-1)\ln(2)\). We then solved for x and found the solution to be \(x = \frac{-\ln(2)}{\ln(3)-\ln(2)}\). To check our solution, we graphed \(y_1 = 3^x\) and \(y_2 = 2^{x-1}\) on a graphing calculator, and confirmed that the point of intersection had the x-value we calculated.

Step 1: Rewrite the Equation with the Same Bases

In order to rewrite the equation with the same bases, we will first take the natural logarithm (ln) of both sides of the equation to bring the exponents down. The equation becomes: \[\ln(3^{x}) = \ln(2^{x-1})\] Now, use the property of logarithms that states \(\ln(a^b) = b\ln(a)\). The equation becomes: \[x\ln(3) = (x-1)\ln(2)\]

Step 2: Solve for x

Next, we need to solve for x by isolating it on one side of the equation. To do this, let's distribute the ln(2) on the right side of the equation and then move all x terms to one side: \[x\ln(3) - x\ln(2) = -\ln(2)\] Now, factor out the x to solve for it: \[x(\ln(3)-\ln(2)) = -\ln(2)\] Finally, divide both sides by \((\ln(3)-\ln(2))\) to find the solution for x: \[x = \frac{-\ln(2)}{\ln(3)-\ln(2)}\]

Step 3: Check the Solution with a Graphing Calculator

Lastly, we will verify our solution using a graphing calculator. In the graphing calculator, graph the following two functions: \(y_1 = 3^x\) \(y_2 = 2^{x-1}\) The point of intersection of the two graphs will give us the value of x, which we calculated above. The calculator should show that the two functions intersect at the x-value that we calculated, confirming that our solution is correct.

Key Concepts

Natural Logarithms

Natural logarithms, commonly noted as \( \ln \), are logarithms with the base \( e \), where \( e \approx 2.71828 \). Using natural logarithms is particularly useful in problems involving exponential equations. This is because they help to simplify expressions involving powers, especially when variables are in the exponent.

In the given problem \( 3^x = 2^{x-1} \), taking the natural logarithm of both sides allows us to "bring down" the exponent using the property \( \ln(a^b) = b \ln(a) \). This makes the equation easier to handle algebraically because it reduces the problem of solving an exponential equation to that of solving a linear equation. This step converts our exponential equation into a form where we can easily isolate and solve for the variable \( x \).

Solving Equations Algebraically

Solving equations algebraically involves manipulation and simplification to find the value of an unknown variable. In the context of our problem, once we take the natural logarithm of both sides, we reach the equation \( x\ln(3) = (x-1)\ln(2) \).

The next step is isolating \( x \) on one side of the equation:

• Distribute \( \ln(2) \) on the right-hand side: \( x \ln(3) - x \ln(2) = -\ln(2) \).
• Factor out \( x \) from the left-hand side: \( x(\ln(3) - \ln(2)) = -\ln(2) \).
• Solve for \( x \) by dividing both sides by \( (\ln(3) - \ln(2)) \): \( x = \frac{-\ln(2)}{\ln(3) - \ln(2)} \).

Each of these steps uses straightforward arithmetic and algebraic manipulation to systematically isolate and solve for \( x \).

Graphing Calculator Verification

Using a graphing calculator is a reliable way to verify solutions we obtain algebraically. For our equation, \( 3^x = 2^{x-1} \), graphing each side of the equation as separate functions will illustrate their point of intersection, providing a visual confirmation of the solution.

To verify:

• Input \( y_1 = 3^x \) and \( y_2 = 2^{x-1} \) into the calculator.
• Observe the graph and find the intersection point of these two curves.
• Check that the \( x \)-coordinate of this intersection matches the solution \( x = \frac{-\ln(2)}{\ln(3) - \ln(2)} \).

This is a great method because it reinforces the idea that our algebraic solution corresponds to what is visually represented in the graph. It provides confidence in the correctness of our calculated value of \( x \).

Logarithm Properties

Understanding logarithm properties is crucial when solving exponential equations. In the problem \( 3^x = 2^{x-1} \), the use of logarithm properties enables us to simplify complex exponential forms.

Some important properties include:

• \( \ln(a^b) = b \ln(a) \): This allows us to move the exponent down in front of the logarithm, which is essential for solving for \( x \).
• \( \ln(a) - \ln(b) = \ln(\frac{a}{b}) \): While not explicitly used in our solution, this property is often valuable in simplifying expressions.
• Changing multiplication to addition: \( \ln(ab) = \ln(a) + \ln(b) \).

These properties help transform complicated exponential equations into simpler, more manageable forms that can be solved using algebraic methods. Knowing and applying these rules effectively is key to mastering exponential problems.