Problem 47 - A Graphical Approach to Precalculus with Limits

Question

Solve each equation. Do not use a calculator. $$27^{4 x}=9^{x+1}$$

Step-by-Step Solution

Answer

x = \frac{1}{5}

Step 1: Express in Terms of Same Base

Start by expressing both sides of the equation in terms of powers of the same base. Notice that both 27 and 9 are powers of 3. Specifically,\[27 = 3^3\]and\[9 = 3^2.\]Thus, you can rewrite the original equation as:\[(3^3)^{4x} = (3^2)^{x+1}.\]

Step 2: Apply the Power of a Power Property

Use the exponentiation rule \((a^m)^n = a^{m imes n}\) to simplify each side of the equation. For the left side,\[(3^3)^{4x} = 3^{12x}.\]For the right side,\[(3^2)^{x+1} = 3^{2(x+1)} = 3^{2x + 2}.\]The equation now becomes:\[3^{12x} = 3^{2x + 2}.\]

Step 3: Equate Exponents

Since the bases are the same, set the exponents equal to each other. This gives us the equation:\[12x = 2x + 2.\]

Step 4: Solve for x

Subtract \(2x\) from both sides of the equation to isolate \(x\):\[12x - 2x = 2.\]This simplifies to:\[10x = 2.\]Finally, divide both sides by 10 to solve for \(x\):\[x = \frac{2}{10} = \frac{1}{5}.\]

Key Concepts

Power of a Power Property

The power of a power property is a special rule in exponentiation that makes dealing with complex exponents much simpler. This property states that when you have an exponent raised to another exponent

• \((a^m)^n\),

it's equivalent to multiplying the two exponents together to get

• \(a^{m \times n}\).

For example, if we have \((3^3)^{4x}\), we use the power of a power property to transform it into \(3^{3 \times 4x} = 3^{12x}\). This simplification is especially useful when solving equations because it reduces the problem to simpler, more manageable terms. Furthermore, applying this property consistently helps in setting up equations that are easier to solve, as seen in the steps where the bases are matched and the equation becomes easier to interpret and solve.

Same Base

Understanding the concept of the same base is crucial when dealing with equations involving exponents. The key insight is recognizing that numbers can often be expressed as powers of a common base. This helps simplify and solve complex equations. In our problem, we were given the equation \(27^{4x} = 9^{x+1}\). Both 27 and 9 can be expressed as powers of 3:

• 27 is \(3^3\)
• 9 is \(3^2\)

By rewriting these numbers as powers of the same base, the equation can transform into \((3^3)^{4x} = (3^2)^{x+1}\). This expression allows us to simplify the problem using exponential properties, turning it into an easier equation with the same base on both sides: \(3^{12x} = 3^{2x + 2}\). Since both sides now share the same base of 3, we can focus on equating their exponents, making the algebraic manipulation straightforward.

Algebraic Equation Solving

Solving algebraic equations involving exponents often requires setting the exponents equal to each other once a common base is identified. In our problem, after simplifying to use the same base, we reached the equation:

• \(12x = 2x + 2\)

The next step is to solve this equation by isolating the variable. To do this:

• Subtract \(2x\) from both sides, which simplifies the equation to \(10x = 2\).
• Divide both sides by 10 to solve for \(x\): \(x = \frac{2}{10} = \frac{1}{5}\).

This systematic approach—simplifying, equating exponents, then using basic algebra—demonstrates the elegance of solving such equations. The solution reveals a clear path to finding the unknown variable, reinforcing the principles behind algebraic problem-solving techniques.