# Condensing And Expanding Logs 3a.2 Homework

## Expanding Log Expressions

The second term above, with just a "5" inside, is as "expanded" as it can get, because there's only just the one thing inside the log. And, because 5 is not a power of 2, there's no simplification I can do. So that part of the expansion is done; I'll just be carrying the "log(5)" along for the ride to the final answer.

In the first term, though, there's still more than just one thing inside the log. In particular, I see that there's an exponent inside the log. However, I can't take the exponent out front yet, because that power is only on the x, not the 8. I have to remember that the rule says that I can only take the exponent out front if it is "on" everything inside the log. So I first need to isolate that part of the argument that has the power on it.

The 8 is multiplied onto the x4, so I can split the factors inside the log by converting to added logs:

log2(8x4) – log2(5) = log2(8) + log2(x4) – log2(5)

Since 8 is a power of 2 (namely, 23), I can simplify the first log to an exact value. Because 23 = 8, then log2(8) = 3, so I get:

log2(8) + log2(x4) – log2(5)

Okay; now I'm finished with the first term, too; I'm only left with the middle term to expand, with the exponent inside its log.

The variable x has the exponent (which is now "on" everything inside its log), so I can use a log rule and move the exponent out in front of the log as a multiplier:

log2(8) + log2(x4) – log2(5)

Each log now finally contains only one thing, and the first log term has been simplified to a numerical value, so this expression is fully expanded. Then my final answer is:

In following my work in the steps of the above computations, you may have felt that I was being a bit confusing, carrying the "log2(5)" and "3" along as I did other steps. But it is important to not drop bits of an exercise as one goes along. In whatever manner you decide to do your work — maybe including doing some of the steps, or at least portions of some of them, off to the side on scratch paper — make sure that all the steps in your final result make sense.

For instance, if I remove the talking between the steps, the previous example is worked out as follows:

Just be careful not to try to do too many things in any one step, at least when you're just getting started.

This is a gawd-awful mess! To do the expansion, I'll be using the log rules, and I'll be taking care not to try to do anything "in my head" or too much all at once.

The first thing I see, inside the log, is that I've got one complicated expression that's divided by another complicated expression. To start my expansion, then, I'll split the division inside the log into subtraction of logs outside.

Inside each of the two log terms I've got now, I find multiplication. So my next step will be to take apart the multiplications inside as addition of logs outside. To make sure I don't mess up my signs, I'll be sure to put grouping symbols around the results of each split.

Now I'll take the middle "minus" through the square brackets.

Inside all but the first term, I find exponents. Since each exponent is "on" the entire contents of its respective log, I can go straight to moving the powers inside to being multipliers outside.

log3(4) + 2 · log3(x – 5) – 4 · log3(x) – 3 · log3(x – 1)

log3(4) + 2log3(x – 5) – 4log3(x) – 3log3(x – 1)

Don't think that this example is "too complicated" to show up on the next test. In fact, you should expect to see at least one question at least this "complicated"!

You can use the Mathway widget below to practice expanding log expressions. Try the entered exercise, or type in your own exercise. Then click the button to compare your answer to Mathway's. (Or skip the widget and continue with the lesson.)

(Clicking on "Tap to view steps" on the widget's answer screen will take you to the Mathway site for a paid upgrade.)

URL: http://www.purplemath.com/modules/logrules2.htm

## Purplemath

Okay; they've told me to "expand", so I know they're wanting me to take this one log apart, into many log terms.

I'll start with the division inside this log. The 5 is divided into the 8x, so I'll apply a log rule and split the numerator and denominator apart by converting the one log with division into two subtracted logs:

= log2(8) + log2(x4) – log2(5)

= 3 + log2(x4) – log2(5)

= 3 + 4log2(x) – log2(5)

The logarithm of a positive number is defined as the power by which another fixed number is raised to get the given number. The fixed number to which the power is raised, is termed as a base. The logarithm is abbreviated as log or sometimes ln.

For example - Logarithm of 10,000 to base 10 will be 4, since 10$^{4}$ = 10,000.

According to the more formal definition -

Let us assume that x = a$^{y}$, then y is supposed to be the logarithm of x at base a. It is expressed in the following way -

log$_{a}$(x) = y

There are two types of bases -

(1) Base 10 - It is termed as common logarithm.

(2) Base e - It is known as natural logarithm, where e is an irrational number which is approximately equal to 2.718.

The two ways to solve the logarithmic problems by using the logarithmic properties. Logarithmic properties are useful for rewrite logarithmic expressions in the simple form by converting complicated products, quotients and exponential forms into simpler sums, differences and products respectively.

Properties:

Property 1: log 1 = 0

Property 2. loga a = 1

Property 3. log(uv) = log u + log v.

Property 4. log$\frac{u}{v}$ = log u - log v.

Property 5. log $u^n$ = n log u.

Many logarithmic expressions may be rewritten, either expanded or condensed, using the above properties.

## Definition

Two very popular methods to solve the logarithmic problems using properties of logarithm to rewrite each expression as a sum, difference or multiple of logarithms. Exponents from the inside of a logarithms and turn them into adding, subtracting or coefficients on the outside of the logarithm. Expanding is breaking down a complicated expression into simpler components and condensing is the reverse of this process.

Expanding Logarithm

Expand log$x^2$y

log$x^2$ y = log$x^2$ + log y = 2 log x + log y

Condensing Logarithm

Condense 2 log x + log y

2 log x + log y = log$x^2$ + log y = log$x^2$ y

## Expanding Logarithms

Expanding of logarithmic problems using properties of logarithms to rewrite each expression as a sum, difference or multiple of logarithms. Turn the exponents from the inside of a logarithms into adding, subtracting or coefficients on the outside of the logarithm.

Properties of expanding logarithms are same as properties of logarithms.

Step for Expanding: (follow properties listed above)

Step 1: Rewrite radicals using rational exponents.

Step 2: Apply Property 3 or 4 to rewrite the logarithm as addition and subtract

Step 3: Apply property 5 to move the exponents out front of the logarithms.

Step 4: Apply property 1 or 2 to simplify the logarithms.

## Condensing Logarithms

Use the properties of logarithms to rewrite each expression as the logarithm of a single quantity. By condense the log, we really mean write it as a single logarithm with coefficient of one using logarithmic properties. When condensing, we always end up with only one log and bring the exponents up.

Properties

1. 0 = log 1

2. 1 = loga a

3. log u + log v = log(uv)

4. log u - log v = log$\frac{u}{v}$

5. n log u = log $u^n$

Step for Condensing Logarithms:

Step 1: Apply Property 5 and move the number in front of the logarithm to the exponent of the variable.

Step 2: Apply Property 3 or 4 to change the addition or subtraction of the logarithms to multiplication or division.

Step 3: Rewrite rational exponents as radicals.

## Expanding Logarithms Examples

Below you could see expanding logarithms examples

### Solved Examples

Question 1: Expand log$\sqrt{xy}$
Solution:

Given log$\sqrt{xy}$

Step 1:

log$\sqrt{xy}$ = log$(xy)^{\frac{1}{2}}$ = $\frac{1}{2}$log xy

using property 5,  log $u^n$ = n log u

Step 2:

$\frac{1}{2}$ log xy = $\frac{1}{2}$ (log x + log y)

using property 3, log(uv) = log u + log v

⇒   log$\sqrt{xy}$ = $\frac{1}{2}$ (log x + log y)

Question 2: Using the properties of logarithms, expand
log$\frac{x^2y}{z}$
Solution:

Given  log$\frac{x^2y}{z}$

Step 1:

log$\frac{x^2y}{z}$ = log$x^2$ y - log z

using property 4, log$\frac{u}{v}$ = log u - log v

Step 2:

log$x^2$ y - log z = log$x^2$ + logy - log z

using property 3, log(uv) = log u + log v

Step 3:

log$x^2$ + logy - log z = 2 log x + logy - log z

using property 5,  log$u^n$ = n log u

⇒   log$\frac{x^2y}{z}$ = 2 log x + logy - log z

Question 3: Using the properties of logarithms,
expand log$\frac{x^5y^2}{z^2}$

Solution:

Given  log$\frac{x^5y^2}{z^2}$

Step 1:

log$\frac{x^5y^2}{z^2}$ = log$x^5y^2$ - log$z^2$

using property 4, log$\frac{u}{v}$ = log u - log v

Step 2:

log$x^5y^2$ - log$z^2$ = log$x^5$ + log$y^2$ - log$z^2$

using property 3, log(uv) = log u + log v

Step 3:

log$x^5$ + log$y^2$ - log$z^2$ = 5 log x + 2 log y - 2 log z

using property 5,  log $u^n$ = n log u

⇒ log$\frac{x^5y^2}{z^2}$ = 5 log x + 2 log y - 2 log z

Question 4: Expand log$\sqrt{x^2 + 1}$ y
Solution:

Given log$\sqrt{x^2 + 1}$ y

Step 1:

log$\sqrt{x^2 + 1}$ y = log$\sqrt{x^2 + 1}$ + log y

using property 3, log(uv) = log u + log v

Step 2:

log$\sqrt{x^2 + 1}$ + log y = $\frac{1}{2}$log($x^2$ + 1) + log y

using property 5,  log $u^n$ = n log u

⇒ log$\sqrt{x^2 + 1}$ y  = $\frac{1}{2}$ log($x^2$ + 1) + log y

## Condensing Logarithms Examples

### Solved Examples

Question 1: Use the properties of logarithms to write as a single logarithm
3 log x + 5 log x - 2 log x
Solution:

Given 3 log x + 5 log x - 2 log x

Step 1:

3 log x + 5 log x - 2 log x = log $x^3$ + log $x^5$ - log $x^2$

using property 5, n log u = log $u^n$

Step 2:

log $x^3$ + log $x^5$ - log $x^2$ = log( $x^3$ . $x^5$) - log $x^2$

using property 3, log u + log v = log(uv)

Step 3:

log( $x^3$ . $x^5$ ) - log $x^2$ = log( $x^8$ ) - log $x^2$

= log$\frac{x^8}{x^2}$

using property 4, log u - log v = log$\frac{u}{v}$

= log $x^6$

⇒ 3 log x + 5 log x - 2 log x = log $x^6$

Question 2: Use the properties of logarithms to write as a single logarithm
5 log x - $\frac{1}{2}$ log y
Solution:

Given  5 log x - $\frac{1}{2}$ log y

Step 1:

5 log x - $\frac{1}{2}$ log y = log $x^5$ - log $y^\frac{1}{2}$

using property 5, n log u = log $u^n$

Step 2:

log $x^5$ - log $y^\frac{1}{2}$ = log $x^5$ - log $\sqrt{y}$

= log $\frac{x^5}{\sqrt{y}}$

using property 4, log u - log v = log$\frac{u}{v}$

⇒ 5 log x - $\frac{1}{2}$ log y = log $\frac{x^5}{\sqrt{y}}$

Question 3: Condense $\frac{1}{2}$ log 2 - $\frac{3}{2}$ log 9 to a single logarithm
Solution:

Given $\frac{1}{2}$ log 2 - $\frac{3}{2}$ log 9

Step 1:

$\frac{1}{2}$ log 2 - $\frac{3}{2}$ log 9 = log $2^\frac{1}{2}$ - log$9^\frac{3}{2}$

using property 5, n log u = log$u^n$

Step 2:

log $2^\frac{1}{2}$ - log $9^\frac{3}{2}$ = log $2^\frac{1}{2}$ - log $3^{2\times\frac{3}{2}}$

= log $\sqrt{2}$ - log $3^3$

= log $\sqrt{2}$ - log 27

= log$\frac{\sqrt{2}}{27}$

using property 4, log u - log v = log$\frac{u}{v}$

⇒ $\frac{1}{2}$ log 2 - $\frac{3}{2}$ log 9 = log$\frac{\sqrt{2}}{27}$

Question 4: Use the properties of logarithms to write as a single logarithm
5 log x - 6 log x + 2 log x
Solution:

Given 5 log x - 6 log x + 2 log x

Step 1:

5 log x - 6 log x + 2 log x = log $x^5$ - log $x^6$ + log $x^2$

using property 5, n log u = log $u^n$

Step 2:

log $x^5$ - log $x^6$ + log $x^2$ = log $x^5$ + log $x^2$ - log $x^6$

= log($x^5 . x^2$) - log $x^6$

using property 3, log u + log v = log(uv)

= log$x^7$ - log $x^6$

Step 3:

log$x^7$ - log $x^6$ = log$\frac{x^7}{x^6}$

using property 4, log u - log v = log$\frac{u}{v}$

= log x

⇒ 5 log x - 6 log x + 2 log x = log x