Expanding and Condensing Logarithmic Expressions

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Objectives
  • Apply multiple logarithm manipulation rules at the same time
  • Begin with a single logarithm of a complicated term and end with sums and differences of several much simpler logarithms (expanding)
  • Begin with a string of sums and differences of logarithms and end with a single logarithm by using logarithm manipulation formulas (condensing)
Lesson Description

Using the logarithmic rules we recently learned, we can now operate on logarithms by either taking a single logarithm of a complex object and breaking it up into simpler logarithms, or conversely, taking several simple logarithms and combining them into a single logarithm. These opposite processes each have their place in your math future, each making certain complex tasks easier.

 
Practice Problems

Practice problems and worksheet coming soon!

 

What Does Expanding and Condensing Mean?

Recently, we learned how to manipulate logarithm expressions by using three "transformations" that helped us re-write logs in a few ways: $$\log(xy) = \log(x) + \log(y)$$$$\log\left(\frac{x}{y}\right) = \log(x) - \log(y)$$$$\log\left(x^a\right) = a\log(x)$$When we studied these transformations, their use may not have been immediately clear. However, in order to perform the tasks that this lesson instructs you how to do, you must remember how to perform these three basic log manipulations.In short, condensing logarithms means taking several log expressions added together and writing them as one concise logarithm. Conversely, expanding a logarithm means taking a single, complicated logarithm and writing it as a sum of simpler logarithms. If you know the three rules above and how to use them, you're ready to rock.

Fully Manipulating Logarithm Expressions

Before we learn a step-by-step approach to expanding and condensing logarithms, let us more clearly define what it means to say that a log is fully expanded or fully condensed.
Define: Condensed LogarithmsA logarithm is fully condensed if:
  • there is one single logarithm
  • the logarithm has a coefficient of 1
For example, $\ln\left(\frac{x^5}{y^2}\right)$ is fully condensed, but $3\ln(x)$ is not.
Define: Expanded LogarithmsGenerally, a logarithm is fully expanded if:
  • every object you are taking the logarithm of is a single variable or number with no exponent
  • every object you are taking the logarithm of is not a fraction
For example:$\log(4) - 4\log(x)$ is fully expanded$\log\left(x^2\right) + \log(y)$ is not fully expanded$\log(y) - \log(x+1) + 3\log(z)$ is fully expanded$2\log\left(\frac{x}{y}\right)$ is not fully expandedTypically, if you are asked to expand a logarithm, you will be given fully condensed log expressions to start with. Similarly if you are asked to condense a logarithm, you will be given fully expanded log expressions to start with. This doesn't have to be true however, nor does it really affect the step by step process.

Expanding Logarithms - Step by Step

In order to work the process clearly, we will use the following example:Example 1$$\log \left(\frac{(5xy)^2}{3w^{-3} z^3}\right)$$Step 1 - Clean Up ArgumentFirst, work inside the log argument to make sure the expression is in a form that will make the rest of your task as easy as possible. Specifically, fix these two issues, if applicable:
  • If there are exponents that apply to the whole argument, distribute them
e.g. $\log\left(\left(xyz\right)^2\right) \longrightarrow$ rewrite as $\log\left(x^2 y^2 z^2\right)$Note that something like$$\log\left(\left(x+y\right)^2\right)$$Does not need any "exponent distribution" because $(x+y)$ is an entire factor, and there are no other factors in this expression. However, something like$$\log\left(\left(\left(x+y\right)^2\left(a-b\right)^3\right)^5\right)$$which has two factors, $(x+y)$ and $(a-b)$, should be cleaned up as$$\log\left(\left(x+y\right)^{10}\left(a-b\right)^{15}\right)$$Example 1 Cont'dIn our example, there is an exponent of $2$ being applied to the entire numerator. We will apply the exponent to each term, in line with this "exponent distribution" idea.$$\log\left(\frac{25x^2 y^2}{3w^{-3} z^3}\right)$$
  • If possible, factor out common terms
e.g. $\log\left(xy+xz\right)$ is better as $\log\left(x\left(y+z\right)\right)$Ultimately we need to work with factors and quotients in the log argument when expanding and contracting, so before proceeding, we should attend to these two algebra manipulations to the extent possible (again, it will make our lives easier for the other steps). Fortunately, the majority of problems will not have common terms that need to be factored out.Example 1 Cont'dOur example does not have any common terms to factor out, so we will simply proceed.Step 2 - Break It UpHaving followed step 1, you have a log with an expression inside that is the product and quotient of factors. Now, using the first and second basic log manipulations,$$\log(xy) = \log(x) + \log(y)$$$$\log\left(\frac{x}{y}\right) = \log(x) - \log(y)$$we can split the logarithm expression into the sum and difference of logs. Everything that was in the numerator will be a positive log. Everything that was in the denominator will be a negative log. Here are some examples:$$\ln(xyz) \longrightarrow \ln(x) + \ln(y) + \ln(z)$$ $$\ln\left(\frac{wx}{yz}\right) = \ln(w) + \ln(x) - \ln(y) - \ln(z)$$ $$\ln\left(\frac{a^2 b^3}{c^4 d^2}\right)$$$$ = \ln\left(a^2\right) + \ln\left(b^3\right) - \ln\left(c^4\right) - \ln\left(d^2\right)$$ Now look at our example:Example 1 Cont'd$$\log\left(\frac{25x^2 y^2}{3w^{-3} z^3}\right)$$$$\longrightarrow \log(25) + \log\left(x\right)^2 - \log(3) -\log\left(w^{-3}\right) -\log\left(z^3\right)$$ Step 3 - ExponentsThe last thing you must do to ensure full expansion is to apply the third log rule$$\ln\left(x^a\right) = a \ln(x)$$and take any exponents that are inside the log and put them in front as a multiplier. E.g.$$\log\left(x^2\right) - \log\left(y^3\right) = 2\log(x) - 3\log(y)$$Now let's finish our example.$$\log(25) + \log\left(x^2\right) - \log(3) -\log\left(w^{-3}\right) -\log\left(z^3\right)$$$$\longrightarrow \log(25) + 2\log(x) -\log(3) + 3\log(w) - 3\log(z)$$Notice how the log with $w$ had a $-3$ exponent, which came down in front and turned the term positive.Here are some examples of expanding a log start to finish. Try them yourself first!
 
Example 2Expand the log entirely.$$\log\left(x^2\right)$$
Show solution
$$\blacktriangleright \,\, 2 \log(x)$$
 
Example 3Expand the log entirely.$$\log\left(\frac{w^2 x^5}{y z^2}\right)$$
Show solution
$\blacktriangleright$ First break up the expression into separate logs based on whether each factor is in the numerator or denominator. Then apply the exponent rule to finish.$$ \log\left(w^2\right) + \log\left(x^5\right) - \log(y) - \log\left( z^2 \right)$$$$ 2 \log(w) + 5 \log(x) - \log(y) - 2 \log(z)$$
 
Example 4Expand the log entirely.$$\log\left(x(x+1)\right)$$
Show solution
$\blacktriangleright$ Make sure you break up separate factors into separate logs, but remember that something like $(x+1)$ is its own entire factor and cannot be further worked on.$$ \log(x) + \log(x+1)$$
 
Example 5Expand the log entirely.$$\ln(x^{-3}) + 2\ln(xy)$$
Show solution
$$\blacktriangleright \,\, -3 \ln(x) + 2 \big[ \ln(x) + \ln(y) \big]$$$$ -3 \ln(x) + 2 \ln(x) + 2 \ln(y)$$

Condensing Logarithms - Step by Step

These steps should seem very similar to doing the expansion steps backward, since we are essentially trying to perform the opposite operation.Here is the example that we will work:Example 6$$4\ln(x) + 5\ln(2) + 2\ln(y) -\ln(4) -3\ln(z+2)$$Step 1 - ExponentsWhatever the expression you're starting out with, the first order of business is to take any constant multipliers in front of each logarithm and rewrite using the exponent log manipulation rule. For example, take something like $3\log{(x)}$ and write is as$$\log\left(x^3\right)$$Essentially, we are now thinking of that third log manipulation rule in reverse from when we used it for expanding logs. The rule works backward and forward - it's the difference between$$\ln\left(x^a\right) = a \ln(x)$$and$$a \ln(x) = \ln\left(x^a\right)$$which is to say, the same thing, only "backwards".Look back to our example.Example 6 Cont'd$$4\ln(x) + 5\ln(2) + 2\ln(y) -\ln(4) -3\ln(z+2)$$$$\longrightarrow \ln\left(x^4\right) + \ln\left(2^5\right) + \ln \left( y^2 \right)- \ln(4) - \ln\left(\left(z+2\right)^3\right)$$Step 2 - Combine Log ExpressionsNow that your log expression has no exponents in front of the individual logarithms, we can combine them into a single log. The mindset we will keep will be similar to what we discussed earlier - just think of positive logarithm terms as belonging to the numerator, and negative logarithm terms belonging to the denominator.Example 6 Cont'd$$\ln\left( \frac{x^4 \cdot 2^5 \cdot y^2}{4 \cdot (z+2)^3} \right)$$$$\longrightarrow \ln\left( \frac{32 x^4 y^2}{4(z+2)^3} \right)$$
Remember!
You can only combine log expressions that are of the same log base. This usually isn't something you'll have to look out for, but advanced or intentionally challenging problems could contain several logs of differing bases. You would probably be expected to simply leave log expressions of different bases separate, in that unusual situation.
Step 3 - SimplifyAt this point you're already very close to done - you have a single logarithm which is the most important property of a condensed logarithm. However, you need to make sure the argument, aka the thing inside the logarithm, is reasonably simplified. If any factors can cancel or combine, that needs to happen. Here's a few examples.$$\log(x) - \log\left(x^2\right)$$$$\longrightarrow \log\left(\frac{x}{x^2}\right)$$$$\longrightarrow \log\left(\frac{1}{x}\right)$$
 
$$\log(36)-\log(2)$$$$\longrightarrow \log\left(\frac{36}{2}\right)$$$$\longrightarrow \log(18)$$Let's return to our working example and finish it off.Example 6 Cont'd$$\ln\left( \frac{32 x^4 y^2}{4(z+2)^3} \right)$$The only simplification we need to do is the constants, which have a common factor.$$\longrightarrow \ln\left( \frac{8 x^4 y^2}{(z+2)^3} \right)$$And that's that!

Put It To The Test

More or less, you should expect to see the same, repetitive types of questions on this when you take a test that explicitly tests this topic - where you're asked to either expand or condense a logarithm. You will need to use this skill from time to time, and you're expected to be able to do it anytime from here on out. So while the topic may be repetitive and tedious to practice, it is worth your time.
 
Example 7Expand the following logarithm.$$\log_3 \left(\frac{m^3p^3}{7nq^4}\right)$$
Show solution
$$\blacktriangleright \,\, \log_3 \left( m^3 \right) + \log_3 \left( p^3 \right) - \log_3 \left( 7 \right) - \log_3 \left( n \right) - \log_3 \left( q^4 \right)$$$$\longrightarrow \,\, 3\log_3 (m) + 3\log_3(p) - \log_3(7) - \log_3(n) - 4\log_3 (q)$$
 
Example 8Expand the following logarithm.$$\log \left(\frac{x+2}{y-3}\right)$$
Show solution
$$\blacktriangleright \,\, \log (x+2) - \log (y-3)$$
 
Example 9Expand the following logarithm.$$\ln \left(\sqrt{12x^9}\right)$$
Show solution
$$\blacktriangleright \,\, \frac{1}{2} \,\, \ln \left( 12x^9 \right)$$$$\longrightarrow \frac{1}{2} \,\, \big[ \ln(12) + \ln\left(x^9\right) \big]$$$$\longrightarrow \frac{1}{2} \,\, \big[ \ln(12) + 9\ln(x) \big]$$$$\longrightarrow \frac{1}{2} \,\, \ln (12) + \frac{9}{2} \ln (x) $$
 
Example 10Expand the following logarithm.$$\log \left(\frac{(a+1)^2 \left(b^3\right)}{\left(c^5 d^6\right)^2}\right)$$
Show solution
$$\blacktriangleright \,\, \log \left(\frac{(a+1)^2 \left(b^3\right)}{c^{10} d^{12}}\right)$$$$\log \left( (a+1)^2 \right) + \log \left( b^3 \right) -\log \left( c^{10} \right) - \log \left( d^{12} \right)$$$$2 \log (a+1) + 3 \log (b) - 10\log (c) - 12 \log (d)$$
 
Example 11Expand the following logarithm.$$\log_5 \left(\frac{5x+w}{2w^3 + xw}\right)$$
Show solution
$\blacktriangleright$ Before we break apart the pieces in the traditional way, it will make our lives better if we first recognize and act on the common factor in the denominator. The rest of the process will be typical.$$\log_5 \left(\frac{5x+w}{w \left( 2w^2 + x\right)}\right)$$$$\log_5 \left( 5x+w \right) - \log_5 (w) - \log_5 \left( 2w^2 + x \right)$$This result is fully simplified, since the sums in each logarithm are entire factors.
 
Example 12Condense the following logarithm.$$\frac{4}{3} \,\, \log_6\left(8x^6\right) - \frac{1}{2} \,\, \log_6\left(10y^4\right)$$
Show solution
$$\blacktriangleright \,\, \log_6 \left( \left( 8x^6 \right)^{4/3} \right) - \log_6 \left( \left( 10y^4\right)^{1/2} \right)$$$$\log_6 \left( 16x^8 \right) - \log_6 \left( y^2 \sqrt{10}\right)$$$$\log_6 \left( \frac{16x^8}{y^2 \sqrt{10}} \right)$$
 
Example 13Condense the following logarithm.$$\log_3 (r) - 5\log_3 (t)$$
Show solution
$$\blacktriangleright \,\, \log_3 (r) - \log_3 \left( t^5\right)$$$$\log_3 \left( \frac{r}{t^5} \right)$$
 
Example 14Condense the following logarithm.$$\frac{\log (x)}{3} + 4 \log (y) + 3 \log \left( z^3 \right)$$
Show solution
$$\blacktriangleright \,\, \log \left( x^{1/3}\right) + \log \left( y^4 \right) + \log \left( z^9 \right)$$$$\log \left( y^4 z^9 \sqrt[3]{x} \right)$$
 
Example 15Condense the following logarithm.$$2\ln (x+1) + 3\ln (y) - 6\ln (y) + \ln(x) - 4\ln (y-1)$$
Show solution
$\blacktriangleright$ Once in a while we'll see a problem that has like terms in it, like this one does. It isn't super important if you don't notice, but it's a shade easier to work the problem if you combine them right off the bat, which we'll do here.$$2\ln (x+1) - 3\ln (y) + \ln(x) - 4\ln (y-1)$$$$\ln \left( (x+1)^2 \right) - \ln \left( y^3 \right) + \ln (x) - \ln \left( (y-1)^4 \right)$$$$\ln \left( \frac{x(x+1)^2}{y^3 (y-1)^4} \right)$$
 
Example 16Condense the following logarithm.$$2 \left( 3\ln (w) - 2\ln (z) \right)$$
Show solution
$\blacktriangleright$ This problem will give you fewer chances to make mistakes if you distribute the $2$ coefficient right away.$$6 \ln (w) - 4 \ln (z)$$$$\ln \left( w^6 \right) - \ln \left( z^4 \right)$$$$\ln \left( \frac{w^6}{z^4} \right)$$
 
Example 17Condense the following into one logarithm.$$\log_2 \left( x^2\right) - \log_2 (4x) - \log_2 (y) + 4$$
Show solution
$\blacktriangleright$ This isn't common, but I have seen it appear on quizzes from time to time. In order to follow our instructions explicitly, we need to re-write $4$ as $\log_2 (16)$.$$\log_2 \left( x^2\right) - \log_2 (4x) - \log_2 (y) + \log_2 (16)$$$$\log_2 \left( \frac{\left(x^2\right)\left( 16 \right)}{ (4x)(y)} \right)$$$$\log_2 \left( \frac{4x}{y} \right)$$
 
Example 18Condense as much as possible.$$\frac{\log(x)}{\log(y)} + 2\log (z) - 3\log (w)$$
Show solution
$\blacktriangleright$ You may occasionally be tempted to combine a ratio of logs, but you cannot. The rules we learned govern the behavior of $\log (x/y)$, not $\log(x) / \log(y)$.$$\frac{\log(x)}{\log(y)} + \log \left( z^2 \right) - \log \left( w^3\right)$$$$\frac{\log(x)}{\log(y)} + \log \left( \frac{z^2}{w^3} \right)$$In short, all we could do was work with the right-most two terms. The first object was a sneaky decoy.
 
Lesson Takeaways
  • Ensure that you understand what properties both fully condensed and fully expanded logarithms have
  • Practice to gain comfortability with the steps for each process, condensing and expanding
  • Know the common tricks that teachers throw at you

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