Solve the equation `3cdotx^(log5^2)+2^(log5^x)=64`.

Answer

To solve the equation \(3 \cdot x^{\log_5 2} + 2^{\log_5 x} = 64\), we can follow these steps:

### Step 1: Rewrite the equation using logarithmic properties
We know that \(a^{\log_b c} = c^{\log_b a}\). We can apply this property to rewrite \(2^{\log_5 x}\):
\[ 2^{\log_5 x} = x^{\log_5 2} \]
Thus, we can rewrite the equation as:
\[ 3 \cdot x^{\log_5 2} + x^{\log_5 2} = 64 \]

### Step 2: Combine like terms
Now, we can factor out \(x^{\log_5 2}\):
\[ (3 + 1) \cdot x^{\log_5 2} = 64 \]
This simplifies to:
\[ 4 \cdot x^{\log_5 2} = 64 \]

### Step 3: Isolate \(x^{\log_5 2}\)
Now, divide both sides by 4:
\[ x^{\log_5 2} = \frac{64}{4} = 16 \]

### Step 4: Rewrite 16 as a power of 2
We know that:
\[ 16 = 2^4 \]
Thus, we have:
\[ x^{\log_5 2} = 2^4 \]

### Step 5: Equate the exponents
Since \(x^{\log_5 2} = 2^4\), we can take logarithm base 5 of both sides:
\[ \log_5 (x^{\log_5 2}) = \log_5 (2^4) \]
Using the property of logarithms, this simplifies to:
\[ \log_5 2 \cdot \log_5 x = 4 \cdot \log_5 2 \]

### Step 6: Solve for \(\log_5 x\)
Assuming \(\log_5 2 \neq 0\), we can divide both sides by \(\log_5 2\):
\[ \log_5 x = 4 \]

### Step 7: Convert back to exponential form
Using the property of logarithms, we can write:
\[ x = 5^4 \]

### Step 8: Calculate the final value of \(x\)
Calculating \(5^4\):
\[ x = 625 \]

Thus, the solution to the equation is:
\[ \boxed{625} \]