How do you find the derivative of #log_(3)x#?

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Answer 1 · 2016-03-15T04:30:51

#frac{"d"}{"d"x}(log_3(x)) = 1/(xln(3))#.

There is an identity that states

#log_a(b) = frac{ln(a)}{ln(b)}#,

for #a > 0# and #b > 0#.

So, we can write

#log_3(x) = ln(x)/ln(3)#

for #x > 0#.

So to find the derivative, it helps if you know that

#frac{"d"}{"d"x}(ln(x)) = 1/x#.

So,

#frac{"d"}{"d"x}(log_3(x)) = frac{"d"}{"d"x}(ln(x)/ln(3))#

#= 1/ln(3) frac{"d"}{"d"x}(ln(x))#

#= 1/(xln(3))#.