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How To Get Rid Of Log In An Equation

Log Bases and Log equations

The Common Logarithm

In chemistry, base 10 is the most important base.  We write

        log x

to mean the log base ten of x.

Example:

        log 10,000,000  =  log 107 =  7

and

        log 0.00000001  =  log 10-8


Example

We can see that

        log 12,343,245

is between 7 and 8 since

        10,000,000  <  12,343,245  <  100,000,000

        log 10,000,000  =  7

and

        log 100,000,000  =  8


Example

We can see that

        log 0.0000145

is between -5 and -4 since

0.00001  <  0.0000145  <  0.0001

and

        log 0.00001  =  -5

and

        log 0.0001  =  -4


Exercise

Use your calculator to find

        log 1,234 3.09

and

        log 0.00234 -2.63


Change of base formula

We next want to be able to use our calculator to evaluate a logarithm of any base.  Since our calculator can only evaluate bases e and 10, we want to be able to change the base to one of these when needed.  The formula below is what we need to accomplish this task.

Proof

We write

        y   =    loga x

So that

        ay   =   x

Take logb of both sides we get

        logb ay =   logb x

Using the power rule:

        y logb a  =  logb x

Dividing by logb a

                   logb x
y  =
logb a

Example

Find

     log2 7

We have

                         log 7
log2 7  =  =  2.807...
log 2


Log Equations

Example

Solve

        log2 x - log2 (x - 2) - 3  =  0

We use the following step by step procedure:

Step 1:  bring all the logs on the same side of the equation and everything else on the other side.

        log2 x - log2(x - 2)  =  3

Step 2: Use the log rules to contract to one log

                    x
log2  =  3
x - 2

Step 3: Exponentiate to cancel the log (run the hook).

               x
   =  23  =  8
x - 2

Step 4: Solve for x

        x  =  8(x - 2)  =  8x - 16

        7x = 16

                 16
x =
7

Step 5: Check your answer

        log2 (16/7) - log2 (16/7 - 2)  =  3


Exercises:

  1. log(x + 2) - log(x - 1) = 1 x = 4/3

  2. log2(x) + log2(x + 5)  = 2 .702


Exponential Equations

Example

Solve for x in

        2x - 1 = 3x + 1

Step 1: Take logs of both sides using one of the given bases

        log2 2x - 1 =  log2 3x + 1

Step 2: Use the log rules to simplify

        x - 1  =  (x + 1) log2 3  =  (x + 1)(log 3)/(log2)  =  1.58(x + 1)

Step 3: Solve for x

        x - 1  =  1.58 x + 1.58

        -.58 x  =  2.58

        x  =  -4.45

Step 4: Check your answer.


Exercises

  1. 3x - 2  =  52x  - 3
    ln(125/9)/ln(25/3)

  2. 21 - x =  3x - 1

    x = 1

For an interactive computer lesson on exponential equations click here

or to play the log memory game click here


Back to the Exponentials and Logs Home Page

Back to the Intermediate Algebra (Math 154) Home Page

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How To Get Rid Of Log In An Equation

Source: http://www.ltcconline.net/greenl/courses/154/logexp/explogeq.htm

Posted by: millerdurstownsee.blogspot.com

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