TAOCP 1.2.2 Exercise 15

Exercise 15. [10] Prove or disprove:

$$\log_b x/y = \log_b x - \log_b y, \quad \text{if} \quad x, y > 0.$$

Verified: yes
Solve time: 29s

The statement is true. Since $x,y>0$, Eq. (11) applies to the product $x\cdot(1/y)$. Therefore

$$ \log_b!\left(\frac{x}{y}\right) = \log_b!\bigl(x(1/y)\bigr) = \log_b x+\log_b(1/y). $$

Also, by Eq. (12),

$$ \log_b!\bigl(y^{-1}\bigr) = (-1)\log_b y = -\log_b y. $$

Substituting this into the previous equation gives

$$ \log_b!\left(\frac{x}{y}\right) = \log_b x-\log_b y. $$

Hence the formula is valid whenever $x,y>0$. ∎