Problem 25 - Precalculus

Question

In Exercises \(1-33,\) solve the equation analytically. $$ 3^{(x-1)}=2^{x} $$

Step-by-Step Solution

Answer

The solution is \(x = \frac{\log(3)}{\log(3) - \log(2)}\).

Step 1: Recognize the Exponential Equation

Notice that we have an equation involving exponents on both sides: \(3^{(x-1)} = 2^{x}\). Our goal is to solve for \(x\). Since the bases are different, we need to transform or simplify the equation to make it solvable.

Step 2: Apply the Logarithm

Taking the logarithm of both sides allows us to bring the exponents down: \(\log(3^{(x-1)}) = \log(2^{x})\). Using the property \(\log(a^b) = b \cdot \log(a)\), we rewrite the equation as \((x-1) \cdot \log(3) = x \cdot \log(2)\).

Step 3: Expand and Rearrange Equation

Distribute \(\log(3)\) on the left-hand side: \(x \cdot \log(3) - \log(3) = x \cdot \log(2)\). Rearrange terms to gather all \(x\)-terms on one side: \(x \cdot \log(3) - x \cdot \log(2) = \log(3)\).

Step 4: Factor and Solve for x

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

Step 5: Simplify the Expression

Calculate \(x\) using the known approximate values of logarithms, or leave it as is for an exact solution. This expression can be evaluated further using a calculator.

Key Concepts

Logarithm Properties

When solving exponential equations, understanding and applying logarithm properties can simplify the process. Logarithms allow us to handle exponents more effectively, especially when the bases are different. The primary property used in these situations is the power rule of logarithms:

• \(\log(a^b) = b \cdot \log(a)\)

This rule helps us rewrite exponential terms so that the variable in the exponent becomes a multiplier in front of the logarithm. In the given equation, applying this property to each side reveals the role of the exponents:

• \(\log(3^{(x-1)}) = (x-1) \cdot \log(3)\)
• \(\log(2^x) = x \cdot \log(2)\)

By transforming exponential terms into linear terms featuring logarithms, you can perform algebraic manipulations more easily. Moreover, logarithms convert multiplicative relationships into additive ones, promoting straightforward solutions by simplifying arithmetic operations.

Exponential Equations

Exponential equations are equations where variables appear in the exponent. They often involve expressions like \(a^x = b^y\), where the goal is to solve for the variable. The complexity arises when the bases of the exponents differ. This is evident in the original exercise:

• \(3^{(x-1)} = 2^x\)

With such equations, direct comparison of exponents is not possible. Therefore, seeking common terms or applying logarithms becomes necessary. The introduced equation requires transforming each side so that they have a similar structure. This approach typically involves logarithms to bring exponents down to a manageable level for further algebraic work. Exponential equations are prevalent across various scientific fields, including physics and finance, making their understanding essential. Approaching them with the right tools, like logarithms, simplifies complex problems and uncovers solutions that might otherwise remain hidden.

Analytical Solution Steps

To solve exponential equations analytically, a clear series of steps should systematically address the transformation and manipulation required to isolate the variable. Here's how the solution steps unfold for the given equation:1. **Identify and Apply Logarithms**: Recognize that both sides of the equation are exponential. Applying logarithms can bring down the exponents to form more accessible types of equations.
2. **Rewrite Using Logarithm Properties**: Use \(\log(a^b) = b\cdot\log(a)\) to express both sides as products involving the exponents:
- \((x-1)\cdot\log(3) = x\cdot\log(2)\)
3. **Distribute and Rearrange**: Separate terms so that all expressions involving \(x\) are isolated on one side:
- Distribute: \(x \cdot \log(3) - \log(3) = x \cdot \log(2)\)
- Rearrange: \(x (\log(3) - \log(2)) = \log(3)\)
4. **Solve for \(x\)**: Factor out \(x\) and solve by dividing through by the coefficient of \(x\):
- \(x = \frac{\log(3)}{\log(3) - \log(2)}\)
The provided solution offers both an exact expression and hints at computational evaluation using log values to further verify and simplify results. This methodical approach ensures clarity and precision in finding the solution.