We denote discount factor
$D(t),$ zero coupon bond
$B(t,T),$ $E_t[X] = E[X|\mathcal{F}(t)]$ and $T$-forward measure $E_t^{T}[\ ].$
First, let me fix the Libor
and Forward Libor
to avoid ambiguity
Libor
$L(t,T):$
$$B(t, T)\cdot \Big(1 + (T-t) L(t, T)\Big) = 1.$$
Forward Libor
$F(t,T-\delta,T):$
$$\Big(1 + (T-t)F(t,T-\delta,T)\Big)B(t,T) = B(t,T-\delta)$$
Now we see the cap
$$C(t;T,L^*) = \dfrac{1}{D(t)}E_t\left[D(T)\delta\Big(F(t,T-\delta,T) - L^*\Big)^+\right]$$
We can change into forward measure
$$C(t;T,L^*) = \delta B(t,T)E^T_t\left[\Big(F(t,T-\delta,T) - L^*\Big)^+\right]$$
and $F(t,T-\delta,T)$ is $T$-forward martingale, the above formula become the standard Black-Scholes.
But if we choose $$C(t;T,L^*) = \dfrac{1}{D(t)}E_t\left[D(T)\delta\Big(L(T-\delta,T) - L^*\Big)^+\right]$$ then we can transform into $$C(t;T,L^*) = (1+\delta L^*)\cdot E^{T}_{t}\left[\left(\dfrac{1}{1+\delta L^*} - B(T-\delta,T)\right)^+\right]$$ it become a bond put option expiring at time $T - \delta$ maturing at time $T.$
But $B(t,T)$ is impossible log-normal
under $T$-forward measure, then we can't use Black-Scholes.
So how to deal with for this case?