In other words, the logarithm gives the exponent as the output if you give it the exponentiation result as the input.

## The Common and Natural Logarithms - Purplemath | Home

. The natural log was defined by equations \eqref{naturalloga} and \eqref{naturallogb}. U shaped relationship definition. \label{lnexpinversesb} \end{gather} These equations simply state that $e^x$ and $\ln x$ are inverse functions. Using the base $b=e$, the product rule for exponentials is \begin{gather*} e^ae^b = e^{a+b} \end{gather*} for any numbers $a$ and $b$. The basic idea A logarithm is the opposite of a power. E relationship to ln. But, since in science, we typically use exponents with base $e$, it's even more natural to use $e$ for the base of the logarithm. Since using base $e$ is so natural to mathematicians, they will sometimes just use the notation $\log x$ instead of $\ln x$.In other words, if we take a logarithm of a number, we undo an exponentiation. In that case, it's good to ask. We'll use equations \eqref{lnexpinversesa} and \eqref{lnexpinversesb} to derive the following rules for the logarithm. The rules apply for any logarithm $\log_b x$, except that you have to replace any occurence of $e$ with the new base $b$. The logarithm with base $b$ is defined so that $$\log_b c = k$$ is the solution to the problem $$b^k=c$$ for any given number $c$ and any base $b$. A logarithm is a function that does all this work for you. Because of this ambiguity, if someone uses $\log x$ without stating the base of the logarithm, you might not know what base they are implying. A dating service. Let's start with simple example. If we plug the value of $k$ from equation \eqref{naturalloga} into equation \eqref{naturallogb}, we determine that a relationship between the natural log and the exponential function is \begin{gather} e^{\ln c} = c.

## Lingnan University - The Liberal Arts …

. The product rule We can use the product rule for exponentiation to derive a corresponding product rule for logarithms