Note that we learned about Exponential Functions here and did some Advanced Factoring with Exponents and Logs in the Advanced Factoring section here. Inverses of Logarithmic Functions can be found in the Inverses of Functions section here.
Introduction to Logarithms (Logs)
What is a Log and Why do we Need Them?
I have to admit that logs are one of my favorite topics in math. I’m not sure exactly why, but you can do so many awesome things with them!
We’ll soon see that Logs can be used to “get the variable in the exponent down” so we can solve for it. But logarithms are also used for many other things, including early on to perform computations – before calculators and computers were around. Have you ever heard of a slide rule? (Ask your parents…or grandparents…)
A slide rule was used (among other things) to multiply and divide large numbers by adding and subtracting their exponents. The numbers on the slide rules had different scales (“logarithmic scales”, meaning that the distance between numbers increase exponentially) and you could simply look up a number, and slide the ruler over to another number to get the number you want. When doing this, you were adding and subtracting exponents, thus multiplying and dividing large numbers. Genius! 
Definition of Logarithm
Remember: A log is in exponent! So when you take the log of something, you are getting back an exponent. The two equations below are two different ways to say the same thing, but the first is an exponential equation, and the second is a logarithmic equation.
Note that $ b$ is called the base of the log, and must be greater than 0 (so we don’t have to deal with complex numbers). Also, the base can’t be 1, or the equations wouldn’t be exponential or logarithmic.
The $ x$ in the log equation is called the argument and it must be greater than 0, again, to avoid complex numbers.
To illustrate how these two equations are related, many times a “loop” is shown, which shows that $ b$ raised to the $ y$ equals $ x$. Again, $ x$ is called the argument of the log, and you can write it as $ {{\log }_{b}}x$ or $ {{\log }_{b}}(x)$. Learn this well!
Note: If there is no $ b$ next to the log, then the base is assumed to be 10; this is called a common logarithm.
Here are some examples of equivalent exponential function and logarithmic functions. See the loop?
Here are some simple log problems where we have to use what we know about exponents to find the log back. You’ll probably have some of these to work on tests without a calculator:
Special Logarithms
Most of the logarithms that you’ll work with have either a base of “10” (because we’ll deal in base 10 with our counting system) or base “$ e$”.
A logarithm with base 10 is called a common logarithm, and when you see “log” without a small subscript for the base, you assume it is base 10. Thus, $ \log \left( 1000 \right)=3$ and $ {{\log }_{10}}\left( 1000 \right)=3$, but we don’t need the 10.
A logarithm with base $ e$ is called the “natural logarithm” and is written as $ \ln \left( x \right)$. Thus, we write $ {{\log }_{e}}\left( x \right)$ as $ \ln \left( x \right)$. Again, the base “$ e$” has many applications in both engineering and economics.
Using Logs (and Exponents) in the Graphing Calculator
You can use graphing calculator keys in the TI-83/84 to find the basic logs: LOG (base 10) and LN. For logs with other bases, you can use a function called LOGBASE under MATH (or ALPHA WINDOW 5), or use what we call a “change of base” formula, that we’ll introduce here and talk about more in Basic Log Properties below.
(We learned how to put exponents in the calculator (using ^) here in the Exponents and Radicals in Algebra section.)
Note that you can also use your calculator to perform logarithmic regressions, using a set of points, like we did here in the Exponential Functions section.
Parent Graphs of Logarithmic Functions
Here are some examples of parent log graphs. I always remember that the “reference point” (or “anchor point“) of a log function is $ (1,0)$ (since this looks like the “lo” in “log”). If the function shifts, this “anchor point” will move. The graph of a log function (a parent function: one that isn’t shifted) has an asymptote of $ x=0$.
When the base is greater than 1 (a growth), the graph increases, and when the base is less than 1 (a decay), the graph decreases. The domain and range are the same for both parent functions.
Remember from Parent Graphs and Transformations that the critical or significant points of the parent logarithmic function $ y={{\log }_{b}}x$ (inverse of exponential function) are $ \displaystyle \left( {\frac{1}{b},-1} \right),\,\left( {1,0} \right),\left( {b,1} \right)$.
Transformations of Log Functions
From Parent Graphs and Transformations, the generic equation for a transformation with vertical stretch $ a$, horizontal shift $ h$, and vertical shift $ k$ is $ f\left( x \right)=a\cdot \log \left( {x-h} \right)+k$ for log functions.
Remember these rules:
- When functions are transformed on the outside of the $ f(x)$ part, you move the function up and down and do “regular” math, as we’ll see in the examples below. These are vertical transformations or translations. Stretches and compressions are performed before shifts.
- When transformations are made on the inside of the $ f(x)$ part, you move the function back and forth (but do the opposite math – basically since if you were to isolate the x, you’d move everything to the other side). These are horizontal transformations or translations.
- When there is a negative sign outside the parentheses, the function is reflected (flipped) across the $ x$-axis; when there is a negative sign inside the parentheses, the function is reflected across the $ y$-axis. Reflections are performed before shifts.
- For log functions, get the new asymptote by setting the log argument to 0 (what’s in parentheses after “log”) and solving for $ x$ (horizontal shift).
- To get the new domain of the log function, set the argument to $ >0$; the domain always changes with the horizontal shift. The range is always $ \left( {-\infty ,\infty } \right)$.
Here are some examples, using t-charts:
Writing Logarithmic Equations from Points and Graphs
You may be also be asked to write log equations, such as the following:
- Write an equation to describe the logarithmic function in form $ y=a{{\log }_{b}}x$, with a given base and a given point.
- Write a logarithmic function in form $ y=a\log \left( {x-h} \right)+k$ from a graph (given asymptote and two points).
Note: We may also be able to use logarithmic regression to find logs equations based on points, like we did here with Exponential Regression.
Let’s try these types of problems:
Basic Log Properties, Including Shortcuts
When working with logs, there are certain shortcuts that you can use over and over again. It’s important to understand these, but later, when using them, be familiar with them, so you can use them quickly. Think of some of the rules as “canceling out” the logs with the exponents. And don’t forget the log “loop”:
Now to the important log properties! These properties are derived from the fact that we add exponents when we multiply terms with exponents, we subtract exponents when we divide, and we multiply exponents when we raise them to a power. These are powerful properties that we’ll need to use to isolate variables in exponents so we can solve for them. You will have to memorize these and remember that for the first three, you must be dealing with logs with the same base:
And don’t forget the basics:
Expanding and Condensing Logs
Now we’ll use these properties to expand and condense logs. Expanding Logs generally means turning the “inside multiplying” (with only one log) to “outside adding” (with multiple logs). Condensing Logs generally means turning the “outside adding” (with multiple logs) to the “inside multiplying” (with only one log). Why do we need to do all this? We will need to expand and condense logs to solve log problems.
Note that when expanding logs, it’s generally a good idea to apply the power rule last (unless whole terms are raised to a power, as in the third example). Also, note that these logs can be written with or without the arguments in parentheses (for example, as $ {{\log }_{3}}5{{x}^{3}}\,\,\,\text{or}\,\,\,\,{{\log }_{3}}\left( 5{{x}^{3}} \right)$, which is different than $ {{\left( {{\log }_{3}}5x \right)}^{3}}$).
Here are examples of expanded logs; assume all log arguments are positive.
Here are examples of condensing logs; assume all log arguments are positive. When condensing logs, it’s generally better to apply the power rule first.
Before getting into solving logs, here’s a crazy type of problem I came across that uses the Change of Base formula to solve:
Solving Exponential and Logarithmic Equations using Logs
Now we can use all these tools to solve exponential and log equations! Remember again that math is just using tools that you have to learn to solve problems.
Here are the basic ways to solve exponential and log problems and examples for each; remember to always check your answer to make sure the argument of logs (what’s directly following the log) is positive!
1. Solving Exponential Equations with Variables in the Exponent: Use the power rule to get the exponent down if the variable is in the exponent (probably the most commonly used “tool”). Before doing this, get the base/exponent by itself and take the log of each side You can use ln, log, or if you have logBASE on the calculator, use the base in the exponential equation.
Here’s a practical example where we can use logs to solve the algebraic equation $ 250=100{{\left( {1.05} \right)}^{t}}$ to get the number of years, $ t$, that it would take to go from $100 to $250 with a yearly interest rate of 5%.
Note: Some teachers just have you use the log “loop” and change of base formula instead. (You could also use the logBASE function in your calculator, without using the change of base.) Do you see the loop?
2. If the same log and same base are on both sides, you can just set arguments of logs equal to each other; this is called the “One-to-One Property of Logarithms”:
3. Use the loop to get the variable out of the log argument. You typically use this if you have logs in the equations. Remember to get the log by itself before you use the “loop”.
4. Or, instead of using the log loop above, add a base to both sides that is the base of the log; if you have an “ln” in the problem, use base “$ e$”. Again, you’ll typically use this when you have logs in the equation. Some of these will look familiar!
5. If $ x$ is underneath a complicated exponent, raise each side to the reciprocal of that exponent. You typically use this if you have variables raised to exponents. (These aren’t technically log problems, but you may see them while studying logs). Be careful though; if an even number is in the numerator of the original fraction, take the positive and negative solutions on the other side.
Solving Log Equations
Let’s do some problems and see what techniques we use:
Here are a few more:
Note: If all else fails, you can solve log equations using a graphing calculator; here’s an example:
Applications of Logs
Again, we typically use logs to solve problems where we have a variable in the exponent; we can use the Power Rule to “get the exponent down”.
We learned in the Exponential Functions section that the following formulas are used for exponential growth and decay, and now we can use logs to solve for variables in the exponents. These are:
Let’s first redo a problem from the Exponential Functions section to see how much easier it is to use logs than “guess and check” when the variable is in the exponent.
Population Growth Problem
Many times problems will give you the exponential formula, and you basically have to plug in to get the answers:
Continuous Compounding Problem:
Using Logs to Find the Rate Problem:
Some problems require a two-part solution, where first we solve for the exponential growth or decay rate – typically the $ k$, and then we solve again, with the $ k$ in the problem.
Revisiting Half Life Problem
We can solve half-life problems using two different methods; we’ll use both methods here. We solved a half-life problem above in the Exponents section, but if you need to find a time (a variable in the exponent), then you need to use logs.
Remember that half-life problems deal with exponential decays that halve for every time period. For example, if we start out with 20 grams, after the next time period, we’d have 10, then 5, and so on. For these problems, the base (decay factor) of the exponential equation is .5.
The trick on half-life problems is to raise the .5 to $ \displaystyle \frac{{\text{time period we want}}}{{\text{time for one half-life}}}$, since this will give us the number of times the substance actually halves.
Solution (Method 1, like we used in the Exponential Functions section):
Solution (Method 2):
Sometimes you will learn to solve a half-life problem using the $ A=P{{e}^{{kt}}}$ formula (which is also seen as $ \displaystyle N\left( t \right)={{N}_{0}}{{e}^{{kt}}}$, the uninhibited growth, or continuous growth formula), and get the $ k$ first, like we did in the flea problem earlier. This method seems a little bit more difficult, but sometimes you are asked for the half-life equation with the $ k$ in it.
Also note that sometimes you are given the equation $ \boldsymbol {A=P{{e}^{{-kt}}}}$ for half-life or any decay problems, and then the $ k$ will end up being positive instead of negative, since there is already a negative sign in the exponent.
Logarithmic Inequalities
You may have to solve inequality problems (either graphically or algebraically) with logarithmic functions. Remember that there is a domain restriction since the argument of a log has to be $ >0$. We saw Exponential Inequalities here. Here are some problems:
Learn these rules, and practice, practice, practice!
For Practice: Use the Mathway widget below to try a Condensing a Log problem. Click on Submit (the blue arrow to the right of the problem) and click on Write as a Single Logarithm to see the answer.
You can also type in your own problem, or click on the three dots in the upper right hand corner and click on “Examples” to drill down by topic.
If you click on Tap to view steps, or Click Here, you can register at Mathway for a free trial, and then upgrade to a paid subscription at any time (to get any type of math problem solved!).
On to Advanced Factoring – you are ready!