SOLUTION: How do I convert the formula L=10* Log(I/Io) into exponent form? I already know how to do the decibels (obv with the formula above) but I can't find the I/Io unless I know how t

Algebra ->  Logarithm Solvers, Trainers and Word Problems -> SOLUTION: How do I convert the formula L=10* Log(I/Io) into exponent form? I already know how to do the decibels (obv with the formula above) but I can't find the I/Io unless I know how t      Log On


   

Question 407091: How do I convert the formula L=10* Log(I/Io) into exponent form?
I already know how to do the decibels (obv with the formula above) but I can't find the I/Io unless I know how to convert that formula into exponent form. Please help :)
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
L=10%2A+log%28%28I%2FI%5B0%5D%29%29
In general, the log%28a%2C+%28p%29%29+=+q is equivalent to p+=+a%5Eq. So you want your equation to have the form of thie general logarithmic equation. This means you want to get the 10 out of the way.

There are two way to get the 10 out of the way:
  • Divide both sides of the equation by 10.
  • Use a property of logarithms, q%2Alog%28a%2C+%28p%29%29+=+log%28a%2C+%28p%5Eq%29%29
I'll do the problem both ways. (They should work out the same.

Dividing by 10:
L%2F10+=+log%28%28I%2FI%5B0%5D%29%29
Now we have the desired form and we can rewrite this in exponential form using the pattern above (and the fact that the base of "log" is 10):
10%5E%28L%2F10%29+=+I%2FI%5B0%5D

Using the property:
L+=+log%28%28%28I%2FI%5B0%5D%29%5E10%29%29
Rewriting in exponential form:
10%5EL+=+%28I%2FI%5B0%5D%29%5E10
To solve this for I%2FI%5B0%5D we find the 10th root of each side:
root%2810%2C+10%5EL%29+=+I%2FI%5B0%5D
root%2810%2C+10%5EL%29 may not look like 10%5E%28L%2F10%29 but they are equal. We can replace the radical in root%2810%2C+10%5EL%29 with and exponent of 1/10:
%2810%5EL%29%5E%281%2F10%29
And the rule for exponents when raising a power to a power is to multiply the exponents:
10%5E%28L%2A%281%2F10%29%29
which simplifies to:
10%5E%28L%2F10%29
So both root%2810%2C+10%5EL%29 and 10%5E%28L%2F10%29 are correct expressions for I%2FI%5B0%5D.