Definition of Logarithm:
\(\text{If and only if } y = a^x ), where ( a > 0 ), ( a \neq 1 ), and ( x > 0 \)
\(\log_a y = x \)
Types of Log:
1. Common log ( \(\log_{10} x \) )
Also written as \(\log x \)
2. Natural log ( \(\log_{2.71828} x \) )
Also written as \(\log_e x \) or \(\ln x \)
Logarithm of 1:
\( \log_a 1 = 0 \)
Logarithm of the Base:
\( \log_a a = 1 \)
\( \log_a a^x = x \)
\( a^{\log_a x} = x \)
\( e^{\ln x} = x \)
\( a^{b\log_a x} = x^b \)
Logarithm of Reciprocal:
\( \log_a\left(\frac{1}{x}\right) = -\log_a x \)
Product Rule:
\( \log_a(x.y) = \log_a x + \log_a y \)
Quotient Rule:
\( \log_a\left(\frac{x}{y}\right) = \log_a x – \log_a y \)
Power Rule:
\( \log_a x^n = n \cdot \log_a x \)
Compression Rule:
\( \log_a(x^m.y^n) = m \cdot \log_a x + n \cdot \log_a y \)
Logarithmic Expansion:
For \( x = a \cdot b \cdot c \)
\( \log_a x = \log_a a + \log_a b + \log_a c \)