Logarithm Formulas

1. \(\log_0 N = \log_1 N =\)  Not Defined
2. \(\log_B 0 =\)  Not Defined
3. \(\log_B 1 =0\)
4. \(\log_B B =1\)
5. If \(\log_B N=P\) then
   1. \(P\) is Real if \(B>0\) (but \(B\neq 1\))   And  \(N>0\). \(P\) is Complex if \(B<0\)  Or  \(N<0\).
   2. \(P\) is Real Positive if Either Both\(B>1\) And \(N>1\) OR if Both   \(0 < B < 1\)  And  \(0 < N < 1\) . \(P\) is Real Negative if only One of   \(0 < B < 1\)  Or  \(0 < N < 1\).
6. Power-Multiplication Rule: \(\log_B A^N = N \log_B A\)
7. Product-Addition Rule: \(\log_B (A_1 \times A_2 \times A_3 ... \times A_N) = \log_B A_1 + \log_B A_2 + \log_B A_3 ... + \log_B A_N\)
8. Division-Subtraction Rule: \(\log_B (\frac{A_1}{A_2}) = \log_B A_1 - \log_B A_2\)
9. Reciprocal Rule:\(\log_B N= \frac{1}{\log_N B}\)
10. Chain Rule: \(\log_A B \times \log_B C = \log_A C\)
11. Using Chain and Reciprocal Rule: \(\log_A C=\frac{\log_B C}{\log_B A}\)
12. Antilog Rule: \(AntiLog_B (\log_B N)=B^{(\log_B N)}= N\)
13. Using Log and AntiLog to Calculate Arbitrary Roots and Powers: \(T^S=B^{(S \times \log_B T)}\)
14. For all Real Values of \(x>0\):
    
    \(\log_e x = \ln x =2[\frac{x-1}{x+1}+\frac{1}{3}{(\frac{x-1}{x+1})}^3+\frac{1}{5}{(\frac{x-1}{x+1})}^5+\frac{1}{7}{(\frac{x-1}{x+1})}^7 + ...]=2 \sum_{n=1}^{\infty} \frac{1}{2n-1}{(\frac{x-1}{x+1})}^{2n-1}\)
15. Since \(e^{i\pi}=-1\), therefore
    1. \(\ln (-1)=i\pi\hspace{.5cm}\Rightarrow \log_B (-1)=\frac{\ln(-1)}{\ln B}=\frac{i\pi}{\ln B}\)
16. If \(N<0\) then
    1. \(\ln (N)=\ln(|N|\times (-1))=\ln(|N|)+\ln(-1)=\ln(|N|) + i\pi\)
       
       \(\Rightarrow \log_B (N)=\frac{\ln N}{\ln B}=\frac{\ln(|N|) + \ln(-1)}{\ln B}=\frac{\ln(|N|) + i\pi}{\ln B}\)
17. If \(N>0\) then
    1. \(\ln (Ni)=\ln N + \ln {(-1)}^{\frac{1}{2}}=\ln N + \frac{1}{2} \ln (-1)= \ln N + \frac{i\pi}{2}\hspace{.5cm}\Rightarrow \log_B (Ni) =\frac{\ln (Ni)}{\ln B}=\frac{\ln N}{\ln B} + \frac{i\pi}{2 \ln B}\)
    2. \(\ln (-Ni)=\ln(N \times -1 \times i)=\ln N + \ln(-1) + \ln i =\ln N + i\pi + \frac{i\pi}{2}=\ln N + \frac{3i\pi}{2}\hspace{.5cm}\Rightarrow \log_B (-Ni)=\frac{\ln (-Ni)}{\ln B}=\frac{\ln N}{\ln B} + \frac{3i\pi}{2 \ln B}\)
18. If \(z\) is a Complex Number such that \(z=a+ ib=re^{i\theta}\) then
    1. \(\ln z=\ln (re^{i\theta})=\ln (r) + \ln(e^{i\theta})=\ln (r) + i\theta\hspace{.5cm}\Rightarrow \log_B z=\frac{\ln z}{\ln B}=\frac{\ln (r) + i\theta}{\ln B}\)