The number of solutions of the equation `2^(x)+2^(x-1)+2^(x-2)=5^(x)+5^(x-1)+5^(x-2)` is equal to

Options

A. `1`
B. `2`
C. `3`
D. none of these

Answer

To solve the equation \( 2^x + 2^{x-1} + 2^{x-2} = 5^x + 5^{x-1} + 5^{x-2} \), we will simplify both sides step by step.

### Step 1: Simplify the left-hand side

The left-hand side can be simplified as follows:

\[ 2^x + 2^{x-1} + 2^{x-2} = 2^x + \frac{2^x}{2} + \frac{2^x}{4} \]

Factoring out \( 2^x \):

\[ = 2^x \left(1 + \frac{1}{2} + \frac{1}{4}\right) \]

Calculating the sum inside the parentheses:

\[ 1 + \frac{1}{2} + \frac{1}{4} = \frac{4}{4} + \frac{2}{4} + \frac{1}{4} = \frac{7}{4} \]

Thus, the left-hand side becomes:

\[ 2^x \cdot \frac{7}{4} \]

### Step 2: Simplify the right-hand side

Now, simplify the right-hand side:

\[ 5^x + 5^{x-1} + 5^{x-2} = 5^x + \frac{5^x}{5} + \frac{5^x}{25} \]

Factoring out \( 5^x \):

\[ = 5^x \left(1 + \frac{1}{5} + \frac{1}{25}\right) \]

Calculating the sum inside the parentheses:

\[ 1 + \frac{1}{5} + \frac{1}{25} = \frac{25}{25} + \frac{5}{25} + \frac{1}{25} = \frac{31}{25} \]

Thus, the right-hand side becomes:

\[ 5^x \cdot \frac{31}{25} \]

### Step 3: Set the simplified sides equal

Now we can set the two simplified sides equal to each other:

\[ 2^x \cdot \frac{7}{4} = 5^x \cdot \frac{31}{25} \]

### Step 4: Rearranging the equation

Rearranging gives:

\[ \frac{2^x}{5^x} = \frac{31}{25} \cdot \frac{4}{7} \]

This simplifies to:

\[ \left(\frac{2}{5}\right)^x = \frac{124}{175} \]

### Step 5: Taking logarithms

Taking logarithms on both sides:

\[ x \log\left(\frac{2}{5}\right) = \log\left(\frac{124}{175}\right) \]

### Step 6: Analyzing the solution

Since \( \log\left(\frac{2}{5}\right) < 0 \) (because \( \frac{2}{5} < 1 \)), the left-hand side will be negative for positive \( x \) and positive for negative \( x \).

Thus, there will be exactly one solution for \( x \) since the logarithm of a positive number is defined and the function is strictly decreasing.

### Conclusion

The number of solutions of the equation is **1**.

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