Find X

Date: 01/22/2001 at 15:44:12
From: Crystal Price
Subject: Find x

How do you do a problem like this?

Use your knowledge of powers and the properties of exponents to find 
the value of x in the equation 8^x + 2 = 2.

From: Doctor Ian
Subject: Re: Find x

Hi Crystal, 

Let's take a look at that equation:

  8^x + 2 = 2

As usual, we want to combine as many numbers as we can, so we can 
subtract 2 from each side of the equation:

  8^x + 2 - 2 = 2 - 2

          8^x = 0

Now we have a problem. Recall that 8^1 = 8, and 8^0 = 1, and 8^(1/n) 
is some root of 8, so it's always greater than zero. 

The problem is that there is NO way that you can raise a positive 
number to any exponent to get zero. So this leads me to believe that 
you really meant to say

  8^(x+2) = 2

When you're using '^' to write exponents that involve more than just a
number, you _need_ to use parentheses.  That is,

    x+2
   8     = 8^(x+2)

   8^x+2 = (8^x) + 2

Anyway, let's look at 

  8^(x+2) = 2

It turns out that 2 is the cube root of 8; that is, 

  2^3 = 8

    2 = 8^(1/3)

So we can use this information:

  8^(x+2) = 2

          = 8^(1/3)

This tells us that 

     x + 2 = 1/3

Why? Just as you can add or subtract the same thing to/from both sides of 
an equation, or multiply both sides by the same thing, you can also 
raise both sides of an equation to the same exponent - or, if both 
sides are the same base raised to an exponent, you can drop the base.  
All that's saying is that if you have 

  a^b = a^c

then you know that b = c.  Does that make sense?

Anyway, if you solve 

  x + 2 = 1/3

you'll get the value of x that makes

  8^(x+2) = 2

a true statement. 

I hope this helps. Let me know if you'd like to talk about this some 
more, or if you have any other questions. 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/