Calcule: (a) $\log _{2} 16$ (b) $\log _{3} 81$ (c) $\log _{5} 125$ (d) $\log _{6} 1296$ (e) $\log _{12} 1728$ (f) $\log _{2} 4096$ (h) $\log _{5}-625$ (i) $\log _{2} \sqrt{2}$ (j) $\log 100$ (k) $\log _{2} 1024$ (I) $\log _{\mathrm{m}}$ m (m) $\log _{1} 16$ (n) $\log _{3}\left(\frac{1}{9}\right)$

Step 1 ：Given the logarithm calculations, we need to find the exponent to which the base must be raised to get the number. This is represented as \(b^y = x\), then \(\log_b x = y\).

Step 2 ：For example, \(\log_2 16 = 4\) because \(2^4 = 16\).

Step 3 ：Let's calculate the logarithms for each given base and number:

Step 4 ：(a) \(\log _{2} 16 = 4\)

Step 5 ：(b) \(\log _{3} 81 = 4\)

Step 6 ：(c) \(\log _{5} 125 = 3\)

Step 7 ：(d) \(\log _{6} 1296 = 4\)

Step 8 ：(e) \(\log _{12} 1728 = 3\)

Step 9 ：(f) \(\log _{2} 4096 = 12\)

Step 10 ：(h) \(\log _{5}-625\) is undefined because the logarithm is only defined for positive numbers different from 1.

Step 11 ：(i) \(\log _{2} \sqrt{2} = 0.5\) because \(\sqrt{x} = x^{0.5}\)

Step 12 ：(j) \(\log 100 = 4.605170185988092\)

Step 13 ：(k) \(\log _{2} 1024 = 10\)

Step 14 ：(l) \(\log _{\mathrm{m}} m = 1\) for any positive number m different from 1

Step 15 ：(m) \(\log _{1} 16\) is undefined because the logarithm is only defined for positive numbers different from 1.

Step 16 ：(n) \(\log _{3}\left(\frac{1}{9}\right) = -2\)

Step 17 ：\(\boxed{\text{Final Answer:}}\)

Step 18 ：(a) \(\log _{2} 16 = 4\)

Step 19 ：(b) \(\log _{3} 81 = 4\)

Step 20 ：(c) \(\log _{5} 125 = 3\)

Step 21 ：(d) \(\log _{6} 1296 = 4\)

Step 22 ：(e) \(\log _{12} 1728 = 3\)

Step 23 ：(f) \(\log _{2} 4096 = 12\)

Step 24 ：(h) \(\log _{5}-625\) is undefined

Step 25 ：(i) \(\log _{2} \sqrt{2} = 0.5\)

Step 26 ：(j) \(\log 100 = 4.605170185988092\)

Step 27 ：(k) \(\log _{2} 1024 = 10\)

Step 28 ：(l) \(\log _{\mathrm{m}} m = 1\) for any positive number m different from 1

Step 29 ：(m) \(\log _{1} 16\) is undefined

Step 30 ：(n) \(\log _{3}\left(\frac{1}{9}\right) = -2\)