SOLUTION: 2/log(base8) a - 4/log(base2) a= 1/log(base 4) a

Question 260582: 2/log(base8) a - 4/log(base2) a= 1/log(base 4) a
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!

With logarithms of all these different bases this problem could be difficult. But we can make this problem easier if we recognize that all the bases are powers of 2. So if we use the change of base formula:

we can rewrite all the logs as base 2 logarithms. This will make the solution much easier. Converting the first log to base 2 we get:

Since , is 3:

Using similar logic on the third log we get:

Substituting these into your equation we get:

Simplifying the fractions within fractions we get:

Now we can eliminate the fractions by multiplying both sides by we get:

All the logs cancel!? This leaves

or

This may seem bizarre but it is a solution. When solving an equation for a variable and the variable disappears like this, then it means that either there is no solution (i.e. there are no numbers the variable can be that make the equation true) or that every possible number can be used for the variable and the equation will always be true. The way to tell which is which is by looking at the equation you get after the variable disappears. If the equation is false then there are no solution. If the equation is true, the all numbers work. Your equation, 2 = 2, is definitely true. So your equation is true for all possible values of a. Since a is the argument of a logarithm, it must be positive. So your solution is all positive numbers.
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