KL divergence to expected value

[hexed plover]

$D_{KL}[a||b] = E_a[log \frac{a}{b}]$

[serene viper]

Richienb

[hexed plover]

$-D_{KL}[a||b] + D_{KL}[a||c] = -E_a[log \frac{a}{b}] + E_a[log \frac{a}{c}]$

[serene viper]

Richienb

[hexed plover]

How do I simplify further to this?

$= E_a[log \frac{b}{c}]$

[serene viper]

Richienb