Warning

This document was partially AI generated, however adapted.

$e$ is used as a base because it has the unique property where the gradient of $y = e^x$ is equal to $e^x$ itself.

Key Definitions

• Definition: $\ln(x)$ is simply shorthand for $\log_{e}(x)$.
• Inverse Property: $e^x$ and $\ln(x)$ are inverse functions. They “undo” each other.
  • $e^{\ln(x)} = x$
  • $\ln(e^x) = x$

Log Laws for $\ln(x)$

1. $\ln(a) + \ln(b) = \ln(ab)$
2. $\ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right)$
3. $\ln(a^k) = k \ln(a)$
4. Crucial Values: $\ln(e) = 1$ and $\ln(1) = 0$.

Solving Equations with $e$ and $\ln$

Example 1: Removing $e$

$e^{2x+3} = 7$

Take $\ln$ of both sides:

$\ln(e^{2x+3}) = \ln(7)$

$2x + 3 = \ln(7)$

$x = \frac{\ln(7) - 3}{2}$

Example 2: Removing $\ln$

$\ln(x - 5) = 2$

Apply $e$ to both sides (exponentiate):

$e^{\ln(x-5)} = e^2$

$x - 5 = e^2$

$x = e^2 + 5$ **