1. Let's start by understanding the number $e = 2.7182818$. 2. The number $e$ is the base of the natural logarithm, which means the logarithm with base $e$ is called the natural logarithm, denoted as $\ln(x)$. 3. For any base $b > 0$ and $b \neq 1$, the logarithm of a number $x$ with base $b$ is defined as $\log_b(x)$, which answers the question: "To what power must we raise $b$ to get $x$?" 4. Since $e$ is a special base, the natural logarithm $\ln(x)$ is defined as $\log_e(x)$. 5. The relationship between logarithms of different bases is given by the change of base formula: $$\log_b(x) = \frac{\ln(x)}{\ln(b)}$$ 6. This means if you want to express $\log_b(x)$ in terms of the natural logarithm, you divide $\ln(x)$ by $\ln(b)$. 7. For example, if $b=10$, then: $$\log_{10}(x) = \frac{\ln(x)}{\ln(10)}$$ 8. The number $e$ is important because the function $f(x) = e^x$ has the unique property that its derivative is itself, and the natural logarithm $\ln(x)$ is its inverse function. Final answer: $e$ is the base of the natural logarithm, and logarithms with any base $b$ can be expressed using $e$ by the formula: $$\log_b(x) = \frac{\ln(x)}{\ln(b)}$$