Exponents and Logarithms

Logarithms and exponents are inverse operations. These properties allow us to simplify, expand, or solve exponential and logarithmic expressions, especially in engineering applications such as signal processing, decay models, and differential equations.

Rule Name	Property
Log of 1	$\log_b (1) = 0$
Log of the same number as base	$\log_b (b) = 1$
Product Rule	$\log_b(m\cdot n) = \log_b (m) + \log_b (n)$
Quotient Rule	$\log_b \left( \frac{m}{n} \right) = \log_b m - \log_b n$
Power Rule	$\log_b (m)^n = n\cdot \log_b (m)$
Change of Base Rule	$\log_a (b) = \frac{\log_c b}{\log_c a}$ (or) $\log_a (b) \cdot \log_c a = \log_c b$
Equality Rule	$\log_a (b) = \log_c (b) \Rightarrow a = c$
Number Raised to Log	$b^{\log_b x} = x$
Other Rules	$\log_a \left( a^m \right) = m$ $- \log_b a = \log_b \left( \frac{1}{a} \right)$ (or) $\log_b \left( \frac{1}{a} \right) = -\log_b a$

Natural Logarithm (ln) and Euler’s Number

Euler’s number $e \approx 2.718$ is the base of the natural logarithm. When working with exponential growth, decay, or continuous compounding, we use:

$$ \ln x = \log_e x $$

This is especially useful when solving equations involving exponential functions like $e^x$, and appears frequently in calculus, differential equations, and engineering analysis.

Common Logarithm (log)

In most scientific and engineering calculators, when you enter log without specifying a base, it is understood to be base 10 by default. This is known as the common logarithm.

$$ \log x \equiv \log_{10} x $$

This is especially useful in engineering fields where base-10 logarithms are standard for expressing quantities that span multiple orders of magnitude, such as decibels (dB), pH values, or Richter scale readings.