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With a logarithmic equation in the form:
log(expression) = number
the next step is to rewrite the equation in exponential form. Since your equation is in this form that is what we will do.
In general is equivalent to . Using this pattern on your equation we get:
Adding 6 we get:
All that's left is to simplify the right side.
If you have trouble with negative and/or fractional exponents it can be helpful to factor the exponent in a special way:
If the exponent is negative, factor out -1.
If the exponent is fractional and the numerator of that fraction is not a 1, then factor out the numerator.
The exponent on 256 is negative so we'll factor out -1 in the exponent:
The exponent is fractional but the numerator is 1. So we are finished factoring the exponent. Each factor of the exponent tells us an operation to perform. The factor of -1 in the exponent tells use that we need to find a reciprocal. The factor of 1/4 tells us to find a 4th root.
And a nice thing about this way of looking at the exponent is that we can do these operations in whatever order we choose. So we'll choose the order that looks easiest. So does it look easier to find a reciprocal of 256 or the reciprocal of 256. A reciprocal is pretty easy but that would mean that we would have to find the 4th root of a fraction. So it 256 is a perfect power of 4 it might be easier to find the 4th root before we introduce a fraction. And 256 does turn out to be a power of 4: . So the 4th root of 256 is 4. Therefore we'll start with the 4th root:
(Note how we remove the corresponding factor after we do its operation.) Then we'll do a reciprocal:
And finally we will check. This is not optional when solving logarithmic equations like yours. You must at least check that all the arguments and bases of all logarithms are valid when the variable is equal to the solution. (Valid arguments and bases are positive. Also, valid bases may not be a 1.) If a "solution" makes any argument or base of any logarithm invalid, then we must reject that "solution".
Use the original equation to check:
Checking A = 6.25:
Simplifying...
At this point we can see that the base is 256 and the argument is 0.25, both valid. So our solution passes the required check.
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