Find the value of the following
`2^(log""_(4)5)`

Answer

To find the value of \( 2^{\log_{4}5} \), we can follow these steps:

### Step 1: Rewrite the logarithm
We know that \( 4 \) can be expressed as \( 2^2 \). Therefore, we can rewrite the logarithm:
\[ \log_{4}5 = \log_{2^2}5 \]

### Step 2: Use the change of base formula
Using the change of base formula for logarithms, we have:
\[ \log_{b^m} a = \frac{1}{m} \log_{b} a \]
Applying this to our logarithm:
\[ \log_{2^2}5 = \frac{1}{2} \log_{2}5 \]

### Step 3: Substitute back into the expression
Now we substitute this back into our original expression:
\[ 2^{\log_{4}5} = 2^{\frac{1}{2} \log_{2}5} \]

### Step 4: Simplify the exponent
Using the property of exponents, we can rewrite this as:
\[ 2^{\frac{1}{2} \log_{2}5} = (2^{\log_{2}5})^{\frac{1}{2}} \]

### Step 5: Evaluate \( 2^{\log_{2}5} \)
By the property of logarithms, we know that \( 2^{\log_{2}5} = 5 \). Therefore:
\[ (2^{\log_{2}5})^{\frac{1}{2}} = 5^{\frac{1}{2}} = \sqrt{5} \]

### Final Answer
Thus, the value of \( 2^{\log_{4}5} \) is:
\[ \sqrt{5} \]

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