Logarithmic Functions

The section covers:

Half-Life problems can be found 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…)

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 y 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 raised to the x equals y.   Again, y 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.

Here are some examples where we change an exponential function to a log function, and a log function to an exponential function.  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.  So  $$\log \left( 1000 \right)=3$$  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 $$ln\left( x \right)$$.  So 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 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.

Parent Graphs of Logarithmic Functions

Here are some examples of parent log graphs.  Again, I always remember that the “anchor point” of the exponential function that we saw earlier is (before any shifting of the graph)  is (0, 1) (since the “e” in “exp” looks round like a “0”),  and the “anchor point” of a log function is (1, 0) (since this looks like  the “lo” in “log”).  We have to also remember that if the function shifts, this “anchor point” will move.  Note also that the graph of an log function (a parent function: one that isn’t shifted) has an asymptote of x = 0.

Notice again that 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.

These parent graphs can be transformed like the other parent graphs in the Parent Functions and Transformations section.

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’re going to 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  $${{\left( {{\log }_{3}}5x \right)}^{3}}$$).

Now let’s go the other way.  When condensing logs, it’s generally better to apply the power rule first.

Solving Exponential Equations using Logs

Now we can use all these tools to solve log equations!  Remember again that math is just using tools that you have to learn to solve problems.

Remember to always check your answer to make sure the argument of logs (what’s directly following the log) is positive!

Let’s just jump in and trying some solving.  There are the basic ways to solve log problems:

1.  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 ln or log of each side.  You’ll typically use this when you don’t have logs in the equation, but have variables in the exponents.  We typically use ln instead of log, unless we’re dealing with base 10.

We can also use the log with the base under the exponent, as in the second case of the first example:

Note:  You might be asked to solve simple problems like these using the log “loop” and change of base formula instead:

2.  If the same log and same base are on both sides, you can just set arguments of logs equal to each other:

3.  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 are not really log problems, but you’ll see them:

4.  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”.

5. Add a base to both sides that is the base of the log; if you have an “ln” in the problem, use base “e“.   You’ll typically use this when you have logs in the equation; you can use this instead of using the log loop.  Some of these will look familiar!

Solving Log Equations

Let’s do some problems and see what techniques we use:

Here are a few more:

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”.

There are basically two formulas used for exponential growth and decay, and when we need to solve for any variables in the exponents, we’ll use logs.  These formulas are   $$A=P{{\left( 1+\frac{r}{n} \right)}^{nt}}$$  and  $$A=P{{e}^{rt}}$$, which is also written in these types of problems as  $$A=P{{e}^{kt}}$$.

Let’s first redo the problem earlier to see how much easier it is to use logs than “guess and check” when the variable is in the exponent.

Problem:

Suppose that a graduating class had 500 students graduating the first year, but after that, the number of students graduating declines by a certain percentage.

(b)  Using this same decay rate, in about how many years will there be less than 300 students?

Solution:

(a)  We got the decay rate (10.6%, or .106) by working the here in the Exponential Functions section.

(b)  Now we can finally use logs instead of “Guess and Check”:

Population Growth Problem

Many times problems will give you the exponential formula, and you basically have to plug in to get the answers:

A population of 50 wolves was introduced into a forest in 2009.   The population is expected to grow by the function $$p\left( t \right)=50{{e}^{.085t}}$$, where t is in years.

(c)  During what year will the population double?

Solution:

We have to be a little careful when given dates like in this problem.  Typically (and unless otherwise stated), t represents the number of years after the first year, in our case, 2009.

Continuous Compounding Problem:

Madison really wants to buy a car.  She needs a down payment of $4000. If she deposits$3500 now with interest compounding continuously at 3%, how many years will it take her to save enough to buy the car?

Solution:

Using Logs to Find the Rate Problem:

Aven wants to buy a car in 4 years and needs a down payment of $2500. If she deposits$2000 now, with interest compounded continuously, what interest rate will she need to get her down payment in time?

Solution:

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.

Problem:

A large colony of fleas is growing exponentially on the family dog (yuck!).  There are 400 flees initially.

Solution:

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.

Problem:

A chemical substance has a half-life of 6 hours.

Solution (Method 2):

Sometimes you will learn to solve a half-life problem using the  $$A=P{{e}^{kt}}$$  formula and get the 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  $$A=P{{e}^{-kt}}$$  for half-life or any decay problems, and then the k will end up being positive instead of negative,since we would already have a negative sign in the exponent.

Transformations, Inverses,  Compositions, and Inequalities of Exponents and Logs

Let’s tie up a few loose ends with Exponential and Logarithmic functions.  We’ve covered Transformations of FunctionsInverses of FunctionsCompositions of Functionsand Radical Inequalities and Quadratic Inequalities in other sections, but let’s specifically focus on exponents and logs.

Transformations of Exponential and Log Functions

We saw that the “anchor point” of an exponential functions (before any shifting of the graph) is (0, 1), the “anchor point” of a log function is (1, 0).  Again, I always remember that the “anchor point” of an exponential function (before any shifting of the graph) is (0, 1) (since the exp looks round like a “0”), and the “anchor point” of a log function is (1, 0) (since this looks more like “log”).  We have to also remember that if the function shiftsthis “anchor point” will move.

Remember that 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.

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.

Here are some examples that we saw in the Parent Graphs and Transformations Section:

Inverses and Compositions of Exponential and Logarithmic Functions

As it turns out, exponential functions are inverses of log functions and of course vice versa!  Let’s show algebraically that the parent exponential and log functions  ($$y={{b}^{x}}\,\,and\,\,y={{\log }_{b}}$$) are inverses – three different ways.  Again, we learned about how to find Inverses in the Inverses of Functions section.

Here are the graphs of the two functions again, so you can see that they are inverses; note symmetry around the line y = x.  Also note that their domains and ranges are reversed:

Let’s find the inverses of the following transformed exponential and log functions by switching the x and the y and solving for the “new” y:

Exponential and Logarithmic Inequalities

You may have to solve inequality problems (either graphically or algebraically) with either exponential or logarithmic functions.  Remember that we learned about using the Sign Chart or Sign Pattern method for inequalities here in the Quadratic Inequalities section, and also we have the domain restriction that the argument of a log has to be > 0.

Let’s do a few inequality problems with exponents and logs:

Learn 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 Solving Inequalities – you are ready!

8 thoughts on “Logarithmic Functions”

1. I am having issues with doing mixed transformations with logarithmic graphs. The usual order doesn’t seem to give the right answer (flipping 1st, stretching/shrinking, then shifting last). In my prof’s example, g(x)=ln(1-x), he shifted first, then reflected. Doing so moves the vertical asymptote to x=1 instead of x=-2 (which is what it is if you reflect 1st then flip). I can’t get an explanation from him, so now I’m confused.

• Thanks for writing! The way I do it is to take a -1 out of the (1-x) to get ln(-(x-1)), so we’d flip first over the y axis and then move 1 unit to the right. Then the asymptote would be at x=1. But if you don’t take the -1 out, you have to shift first, and then reflect. Does that make sense? Also, graph it on the graphing calculator to see it. Lisa

2. Hi Lisa,

First, thank you for creating such an amazing website, and explaining these concept in such a user-friendly manner. It is extremely helpful.

I noticed something in this chapter. In your expanding and condensing logs section, the example seems to contain a typo, in the second last line, the 1/4 that is outside the brackets seems to have applied to the last term (2log…) although this term is still inside the brackets. Could you take a quick look please?

Thanks again. (:

• Thanks so much for finding this – boy, you are good! Please let me know if you see anything else, and spread the word about She Loves Math 😉 Lisa