# 3.4: Algebra Tips and Tricks Part VI (Logarithms)

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## 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*}

This page titled 3.4: Algebra Tips and Tricks Part VI (Logarithms) is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Tyler Seacrest via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.