Question 407091
{{{L=10* log((I/I[0]))}}}
In general, the {{{log(a, (p)) = q}}} is equivalent to {{{p = a^q}}}. 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.<br>
There are two way to get the 10 out of the way:<ul><li>Divide both sides of the equation by 10.</li><li>Use a property of logarithms, {{{q*log(a, (p)) = log(a, (p^q))}}}</li></ul>I'll do the problem both ways. (They should work out the same.<br>
Dividing by 10:
{{{L/10 = log((I/I[0]))}}}
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^(L/10) = I/I[0]}}}<br>
Using the property:
{{{L = log(((I/I[0])^10))}}}
Rewriting in exponential form:
{{{10^L = (I/I[0])^10}}}
To solve this for {{{I/I[0]}}} we find the 10th root of each side:
{{{root(10, 10^L) = I/I[0]}}}
{{{root(10, 10^L)}}} may not look like {{{10^(L/10)}}} but they are equal. We can replace the radical in {{{root(10, 10^L)}}} with and exponent of 1/10:
{{{(10^L)^(1/10)}}}
And the rule for exponents when raising a power to a power is to multiply the exponents:
{{{10^(L*(1/10))}}}
which simplifies to:
{{{10^(L/10)}}}
So both {{{root(10, 10^L)}}} and {{{10^(L/10)}}} are correct expressions for {{{I/I[0]}}}.