3.4: Algebra Tips and Tricks Part VI (Logarithms)

Logarithms

A logarithm is the inverse function to an exponential function. For example, for the exponential function $y = 2^x$, if we have an input of $x = 6$, we get an output of $y = 64$, and we write $64 = 2^6$. The logarithmic function $y = \log_2(x)$ is the reverse of this. We swap the input and the output, so now $x = 64$ and $y = 6$. We see $6 = \log_2(64)$.

In calculus, we will mostly use the exponential function $e^x$ and its inverse, $\ln(x)$. Below are some important formulas:

$$\begin{align*} e^{\ln(x)} & = x \\ \ln(e^x) & = x \\ \ln(x) + \ln(y) & = \ln(xy) \\ \ln(x) - \ln(y) & = \ln\left(\frac{x}{y}\right) \\ a \ln(x) & = \ln(x^a) \end{align*}$$

Examples:

$\ln(x^2) - \ln(x)$.

There are two ways to do this one. First, we can bring down the exponent of two down in front $\ln(x^2) = 2 \ln(x)$. Then can combine the like terms of $2\ln(x)$ and $\ln(x)$:

$$\begin{align*} \ln(x^2) - \ln(x) & = 2 \ln(x) - \ln(x) \\ & = \boxed{\ln(x)} \end{align*}$$

Alternatively, we can rewrite the subtraction as a division, like so:

$$\begin{align*} \ln(x^2) - \ln(x) & = \ln\left(\frac{x^2}{x}\right) \\ & = \boxed{\ln(x)} \end{align*}$$

Either way we get the same answer!

$\ln(e^3 x^4) - 3 \ln(x)$.

First, we rewrite the multiplication using addition. Then we can simply from there.

$$\begin{align*} \ln(e^3 x^4) - 3 \ln(x) & = \ln(e^3) + \ln(x^4) - 3 \ln(x)\\ & = 3 + 4 \ln(x) - 3 \ln(x) \\ & = \boxed{3 + \ln(x)} \end{align*}$$

$\ln(\sqrt{x})$.

We know , so .

$\ln\left(\frac{\sqrt{x} y}{z^3}\right) - \ln\left(\frac{z}{\sqrt{x} y^3}\right)$.

We can rewrite all the products and divisions as addition and subtraction:

$$\begin{align*} \ln\left(\frac{\sqrt{x} y}{z^3}\right) - \ln\left(\frac{z}{\sqrt{x} y^3}\right) & = \ln(\sqrt{x}) + \ln(y) - \ln(z^3) - [\ln(z) - \ln(\sqrt{x}) - \ln(y^3)] \\ & = \frac{1}{2} \ln(x) + \ln(y) - 3 \ln(z) - \ln(z) + \frac{1}{2} \ln(x) + 3 \ln(y) \\ & = \boxed{\ln(x) + 4 \ln(y) - 4 \ln(z)}. \end{align*}$$