Question 448377
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.