Convert the given logarithm to exponential form. For example, the log sqrt(2) (12) = x would be expressed in exponential form as sqrt(2)^x = 12.
Take the natural logarithm, or logarithm with base 10, of both sides of the newly formed exponential equation.
log( sqrt(2)^x) = log (12)
Using one of the properties of logarithms, move the exponent variable to the front of the equation. Any exponential logarithm of the type log a ( b^x) with a particular "base a" can be rewritten as x*log a (b). This property will remove the unknown variable from the exponent positions, thereby making the problem much easier to solve. In the previous example, the equation would now be written as:
x*log(sqrt(2)) = log (12)
Solve for the unknown variable. Divide each side by the log(sqrt(2)) to solve for x:
x=log(12)/log(sqrt(2))
Plug this expression into a scientific calculator to get the final answer. Using a calculator to solve the example problem gives the final result as x = 7.2 .
Check the answer by raising the base value to the newly calculated exponential value. The sqrt(2) raised to a power of 7.2 results in the original value of 11.9, or 12. Therefore, the calculation was done correctly:
sqrt(2)^7.2 =11.9