how to evaluate logs with roots
how to evaluate logs with roots
There are two things that will help when simplifying logarithms that contain roots:
First recall that the square root of a number is the same as raising that number to the 1/2 power.
√a = a ^{1/2}
And second, that the log of a number raised to a power is equal to the power multiplied by the log of the number (not raised to any power).
log a ^{b} = b log a
It may also help to recall that:
log ab = log a + log b (note: sliderules, the devices used before electronic calculators, are based on this equation)
And to complete the simplification you should rationalize the denominator by multiplying both the numerator and denominator by root 2
Here are the steps carried out:
log (s√7) / (t√2)
=( log s + log √7) /( t√2)
= (log s + log 7 ^{1/2} ) / t√2)
= ( log s + (1/2) log 7) / (t√2)
= [ ( log s + (1/2) log 7) / (t√2) ] (√2) / (√2)
= (√2) ( log s + (1/2) log 7) / 2t
Hope this helps.