To find the value of [tex]\(\log_4 \left( 4^{\frac{1}{5}} \right)\)[/tex], let's follow a step-by-step approach using the properties of logarithms.
1. Identify the problem: We need to find the logarithm of a number to a given base.
[tex]\[
\log_4 \left( 4^{\frac{1}{5}} \right)
\][/tex]
2. Recall the logarithm property: One of the key properties of logarithms is that if the base and the number inside the logarithm are the same, i.e., [tex]\(\log_b (b^x)\)[/tex], it simplifies to [tex]\(x\)[/tex].
3. Apply the property: Use the property [tex]\(\log_b (b^x) = x\)[/tex] to our expression. Here, the base [tex]\(b\)[/tex] is 4, and the exponent [tex]\(x\)[/tex] is [tex]\(\frac{1}{5}\)[/tex].
[tex]\[
\log_4 \left( 4^{\frac{1}{5}} \right) = \frac{1}{5}
\][/tex]
4. Simplification: This directly simplifies the given logarithmic expression to the exponent.
[tex]\[
\log_4 \left( 4^{\frac{1}{5}} \right) = \frac{1}{5}
\][/tex]
5. Convert to decimal form: The fraction [tex]\(\frac{1}{5}\)[/tex] is equivalently written in decimal form as 0.2.
[tex]\[
\log_4 \left( 4^{\frac{1}{5}} \right) = 0.2
\][/tex]
Therefore, the value of [tex]\(\log_4 \left( 4^{\frac{1}{5}} \right)\)[/tex] is [tex]\(\boxed{0.2}\)[/tex].
