Solve the following equations.

1. Rewrite the equation using laws of indices to identify a common base term

$5^x - 5^{\frac{x}{2}} \cdot 5^1 = 6$
$(5^{\frac{x}{2}})^2 - 5(5^{\frac{x}{2}}) - 6 = 0$

2. Use substitution to form and solve a quadratic equation

Let $y = 5^{\frac{x}{2}}$.
$y^2 - 5y - 6 = 0$
$(y - 6)(y + 1) = 0$
$y = 6 \quad \text{or} \quad y = -1$

3. Solve for $x$ by taking logarithms and reject the invalid solution

$5^{\frac{x}{2}} = 6 \quad \text{or} \quad 5^{\frac{x}{2}} = -1 \text{ (rejected since } 5^{\frac{x}{2}} > 0)$
Taking $\ln$ on both sides for the valid equation:
$\ln(5^{\frac{x}{2}}) = \ln 6$
$\frac{x}{2} \ln 5 = \ln 6$
$x = \frac{2 \ln 6}{\ln 5}$
$x \approx 2.2266 \approx 2.23 \text{ (to 3 s.f.)}$