Logarithms

At a glance 13 formulas

1. 1Definition$\log_a x = y \iff a^y = x$$a > 0,\ a \ne 1,\ x > 0$
2. 2Values to know$\log_a 1 = 0,\quad \log_a a = 1$$a^{\log_a x} = x$
3. 3Product and power$\log mn = \log m + \log n$$\log m^{k} = k\log m$
4. 4Quotient$\log \dfrac{m}{n} = \log m - \log n$$\log (m + n) \ne \log m + \log n$
5. 5Change of base$\log_a b = \dfrac{\log b}{\log a}$$\log_a b \times \log_b a = 1$
6. 6Power in the base$\log_{a^k} x = \dfrac{1}{k}\,\log_a x$$\log_{a^k} x^{m} = \dfrac{m}{k}\,\log_a x$
7. 7Chain of logs$\log_a b \times \log_b c \times \log_c d = \log_a d$
8. 8The flip$a^{\log_b c} = c^{\log_b a}$
9. 9Nested logsPeel from the outside with $\log_a y = k \Rightarrow y = a^k$: $\log_2(\log_3 y) = 1$ gives $\log_3 y = 2$, so $y = 9$.
10. 10Check the domainEvery log needs a positive input and a positive base other than 1. Solve first, then throw out any root that makes an input 0 or negative.
11. 11Log inequality$a > 1:\ \log_a x > \log_a y \iff x > y > 0$$0 < a < 1:\ \log_a x > \log_a y \iff 0 < x < y$
12. 12Quadratic in a logWith $\log_a x$ and $\log_x a$ together, put $t = \log_a x$, so $\log_x a = \frac{1}{t}$. If the roots are $t_1, t_2$, the solutions $x$ multiply to $a^{t_1 + t_2}$.
13. 13Number of digits$\text{digits in } N = \lfloor \log_{10} N \rfloor + 1$$\text{whole } N \ge 1;\ \text{e.g. } \log_{10} 2^{50} \approx 15.05,\ \text{so 16 digits}$