Note that **Exponential and Logarithmic Differentiation** is covered here.

**Exponential and Logarithmic Differentiation** and** Integration** have a lot of practical applications and are handled a little differently than we are used to. For a review of these functions, visit the **Exponential Functions** section and the **Logarithmic Functions** section.

## Introduction to Exponential and Logarithmic Differentiation and Integration

Before getting started, here is a table of the most common **Exponential and Logarithmic formulas **for **Differentiation **and** Integration**:

Actually, when we take the integrals of exponential and logarithmic functions, we’ll be using a lot of **U-Substitution Integration**, so you may want to review it.

## Review of Logarithms

When we learned the Power Rule for Integration here in the **Antiderivatives and Integration** section, we noticed that if $ n=-1$, the rule doesn’t apply since the denominator would be $ 0$: $ \displaystyle \int{{{{x}^{n}}}}dx=\frac{{{{x}^{{n+1}}}}}{{n+1}}\,+C,\,\,n\ne 1$. So, when we try to integrate a function like $ \displaystyle f\left( x \right)=\frac{1}{x}={{x}^{{-1}}}$, we have to do something “special”; namely learn that this integral is $ \ln \left( x \right)$. Remember that $ \ln x$ is the same as $ {{\log }_{e}}x$, where $ e\approx 2.718$ (“$ e$” is **Euler’s Number**). A log is the exponent raised to the base power ($ a$) to get the argument ($ x$) of the log (if “$ a$” is missing, we assume it’s **10**).

Here are some logarithmic properties that we learned here in the **Logarithmic Functions** section; note we could use $ {{\log }_{a}}x$ instead of $ \ln x$.

## The Log Rule for Integration

We learned that the differentiation rule for log functions is $ \displaystyle \frac{d}{{dx}}\left[ {\ln u} \right]du=\frac{{{u}’}}{u}$.

From this, we can get the **Log Rules for Integration**; you’ll probably just want to memorize these. Remember that | | is the **Absolute Value Function**, which means always take the positive of what’s inside. The reason we use an absolute value is that the natural logarithm function is only defined for $ x>0$.

Most of these problems involve **U-Substitution Integration** and some require doing **Polynomial Long Division** before integrating when the degree (largest exponent of all the terms) of the numerator is greater than the degree of the denominator.

Here are some **Logarithmic Integration** problems; assume that $ x$ cannot be equal to any value that makes a denominator **0**:

## Integrals of Trigonometric Functions using “ln”

We learned integrals of some of the trig functions here in the **Antiderivatives and Indefinite Integration** section (included below), but now that we know some log rules, we’ll introduce the rest of the trig integrals. The new trig integrals may be proved by using the log integration rules, but you’ll probably just want to memorize these:

Here are some **Trig Integration** problems; notice that sometimes we can’t use the above equations, but have to work with the log integral rules:

## Integrals of $ \boldsymbol {{{e}^{u}}}$ and $ \boldsymbol {{{a}^{u}}}$

The integrals of the exponential functions $ {{e}^{u}}$ and $ {{a}^{u}}$ mainly involves **U-Substitution**. When we take the integral of a base other than $ e$, we can either convert the function to base $ e$ using the formula $ {{a}^{x}}={{e}^{{\left( {\ln a} \right)x}}}$ (since $ \displaystyle {{e}^{{\left( {\ln a} \right)x}}}={{\left( {{{e}^{{\left( {\ln a} \right)}}}} \right)}^{x}}={{a}^{x}}$), or remember the formula below:

**Integrals of **$ \boldsymbol {{{e}^{u}}}$** and **$ \boldsymbol {{{a}^{u}}}$

$ \int{{{{e}^{u}}}}={{e}^{u}}+C$ $ \displaystyle \int{{{{a}^{u}}}}du=\left( {\frac{1}{{\ln a}}} \right){{a}^{u}}+C$

Here are some **Exponential Integration** problems. It’s a good idea differentiate back to make sure you got the right answer!

First, the $ \boldsymbol {{{e}^{u}}}$** integration** problems. Note that if there are **multiple instances** of $ {{e}^{x}}$ in the problem, we usually include $ {{e}^{x}}$ in the $ u$ part of the problem; otherwise we don’t.

And now, the $ \boldsymbol {{{a}^{u}}}$** integration** problems:

**Understand these rules and practice, practice, practice!**

Click on Submit (the arrow to the right of the problem) to solve this problem. You can also type in more problems, or click on the 3 dots in the upper right hand corner to drill down for example problems.

If you click on “Tap to view steps”, you will go to the **Mathway** site, where you can register for the **full version** (steps included) of the software. You can even get math worksheets.

You can also go to the **Mathway** site here, where you can register, or just use the software for free without the detailed solutions. There is even a Mathway App for your mobile device. Enjoy!

**On to** **Exponential Growth Using Calculus – you’re ready!**