if [2^x+(22)^x+(222)^x]/[3^x+(33)^x+(333)^x] =9/4 then x equals

To solve the equation \(\frac{2^x + 22^x + 222^x}{3^x + 33^x + 333^x} = \frac{9}{4}\), we can approach it step by step. This is a ratio of exponential functions, which often means we can simplify or manipulate the equation to find the value of \(x\).

Rearranging the Equation

First, let's clear the fraction by cross-multiplying:

4(2^x + 22^x + 222^x) = 9(3^x + 33^x + 333^x)

Expanding Both Sides

This gives us two sides to work with:

• Left Side: \(4 \cdot 2^x + 4 \cdot 22^x + 4 \cdot 222^x\)
• Right Side: \(9 \cdot 3^x + 9 \cdot 33^x + 9 \cdot 333^x\)

Identifying Patterns

Next, let's examine the terms in the equation. Notice that the bases on both sides are multiples of each other:

• 2, 22, and 222 can be seen as \(2 \cdot 1\), \(2 \cdot 11\), and \(2 \cdot 111\).
• 3, 33, and 333 can be viewed as \(3 \cdot 1\), \(3 \cdot 11\), and \(3 \cdot 111\).

This suggests that we might be able to express the terms in a more manageable form if we factor out common bases.

Factoring Out Common Bases

Let's factor out \(2^x\) from the left side and \(3^x\) from the right side:

4 \cdot 2^x \left(1 + \left(\frac{22}{2}\right)^x + \left(\frac{222}{2}\right)^x\right) = 9 \cdot 3^x \left(1 + \left(\frac{33}{3}\right)^x + \left(\frac{333}{3}\right)^x\right)

Now, simplifying the terms gives us:

4 \cdot 2^x \left(1 + 11^x + 111^x\right) = 9 \cdot 3^x \left(1 + 11^x + 111^x\right)

Dividing Both Sides

If we assume \(1 + 11^x + 111^x \neq 0\), we can divide both sides by this expression:

4 \cdot 2^x = 9 \cdot 3^x

Rearranging for a Single Exponential Equation

Now we can rearrange this equation:

\frac{2^x}{3^x} = \frac{9}{4}

This can be rewritten using properties of exponents:

\left(\frac{2}{3}\right)^x = \frac{9}{4}

Taking Logarithms

To solve for \(x\), take the logarithm of both sides:

x \cdot \log\left(\frac{2}{3}\right) = \log\left(\frac{9}{4}\right)

Solving for \(x\) gives us:

x = \frac{\log\left(\frac{9}{4}\right)}{\log\left(\frac{2}{3}\right)}

Calculating the Value

Using known logarithmic values:

\log\left(9\right) = 2 \cdot \log\left(3\right) \quad \text{and} \quad \log\left(4\right) = 2 \cdot \log\left(2\right)

Substituting these values gives:

x = \frac{2 \cdot \log(3) - 2 \cdot \log(2)}{\log(2) - \log(3)}

Therefore, through careful manipulation of the given exponential equation, we find the value of \(x\). This approach not only provides the solution but also reinforces the understanding of how to manipulate exponential functions and logarithms effectively.