Represent variables as powers of 5

We are given $a = 5^{0.2}$ and $b = 5^{1.3}$.

$a = 5^{0.2}, \quad b = 5^{1.3}$

Substitute into the relation

Substitute the expressions for $a$ and $b$ into the given equation $a^k = b$.

$(5^{0.2})^k = 5^{1.3}$

Apply exponent power rule

Using the rule $(x^m)^n = x^{mn}$, we simplify the left side to $5^{0.2k} = 5^{1.3}$.

$5^{0.2k} = 5^{1.3}$

Solve for k

Since the bases are equal, we equate the exponents: $0.2k = 1.3$. Dividing both sides by 0.2 gives $k = 6.5$.

$k = \frac{1.3}{0.2} = 6.5$