SOLUTION: solve for x: 2^logx= 1/4

Question 448377: solve for x:
2^logx= 1/4
Answer by bucky(2189) (Show Source): You can put this solution on YOUR website!
If you have a problem in which a variable appears in an exponent, one of the things to consider is taking a logarithm of both sides. This problem is:
2^(logx) = 1/4
Since the variable x is in the exponent, take the log of both sides:
log[2^(logx)] = log(1/4)
One of the properties of logarithms is that exponents come out as multipliers. This rule applies to the left side of this equation. Converting the exponent (log 2x) to a multiplier makes the equation become:
log(2)*logx = log(1/4)
Note that log(2) and log(1/4) are just numbers that can be found using a calculator. Log(2) = 0.301029995 and log(1/4) = log(0.25) = -0.602059991
Substituting these for log(2) and log(1/4) makes the equation become:
(0.301029995)*logx = -0.602059991
Divide both sides of this equation by 0.301029995 and the equation reduces to:
logx = -2
Ten is the base of the term "log". By definition "if log to the base 10 of x equals y this is equivalent to saying the base 10 raised to the exponent y equals x." Think of this and become familiar with this form of conversion.
As this definition applies to the equation:
logx = -2
We can say that 10 (the base) raised to the exponent -2 equals x. In equation form this is:
x = 10^(-2)
But by the rules of exponents 10^(-2) = 1/(10^2) = 1/100 = 0.01
This results in the answer:
x = 0.01
You can check this by substituting 0.01 for x in the original problem:
2^(log(x))
Substituting 0.01 for x:
2^(log(0.01))
Use a calculator to find that the log(0.01) = -2
So our term above becomes:
2^(-2)
By the rules of exponents, a negative exponent is equivalent to 1 over the term with a positive exponent. So this becomes:
2^(-2) = 1/[2^2) = 1/4
and this is exactly as the original problem statement said it should be.
When x equals 0.01, the term 2^logx does equal 1/4 or 0.25
Hope this helps you to understand the problem and how to think your way through it.

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