how to evaluate logs with roots

how to evaluate logs with roots

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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.