How do you find the derivative of ${log}_{3} x$ $log_(3)x$ ?

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How do you find the derivative of ${log}_{3} x$ $log_(3)x$ ?

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Answer 1 · 2016-03-15T04:30:51

$\frac{d}{d x} ({log}_{3} (x)) = \frac{1}{x ln (3)}$ $frac{"d"}{"d"x}(log_3(x)) = 1/(xln(3))$ .

Explanation:

There is an identity that states

${log}_{a} (b) = \frac{ln (a)}{ln (b)}$ $log_a(b) = frac{ln(a)}{ln(b)}$ ,

for $a > 0$ $a > 0$ and $b > 0$ $b > 0$ .

So, we can write

${log}_{3} (x) = \frac{ln (x)}{ln (3)}$ $log_3(x) = ln(x)/ln(3)$

for $x > 0$ $x > 0$ .

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

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

So,

$\frac{d}{d x} ({log}_{3} (x)) = \frac{d}{d x} (\frac{ln (x)}{ln (3)})$ $frac{"d"}{"d"x}(log_3(x)) = frac{"d"}{"d"x}(ln(x)/ln(3))$

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

$= \frac{1}{x ln (3)}$ $= 1/(xln(3))$ .

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Explanation:

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