# Derivative of log_{a} x

How to find the derivative of the given function by using the derivative of log_{a} x: formulas, 2 examples, and their solutions.

## FormulaDerivative of log_{a} x

The derivative of log_{a} x, [log_{a} x]',

is 1/(x ln a).

## Exampley = log_{2} (x^{3} - 8x), y' = ?

y = log_{2} (x^{3} - 8x)

is a composite function of

y = log_{2} (whole) and (whole) = 2x + 7.

So the derivative of the composite function is,

the derivative of the outer function log_{2} (x^{3} - 8x), 1/[(x^{3} - 8x) ln 2]

times,

the derivative of the inner function x^{3} - 8x, 3x^{2} - 8.

Arrange the expression.

Then

(3x^{2} - 8)/[(x^{3} - 8x) ln 2]

is the derivative of log_{2} (x^{3} - 8x).

## FormulaDerivative of log_{a} |x|

The derivative of log_{a} |x|, [log_{a} |x|]',

is also 1/(x ln a).

## Exampley = log_{5} 5x^{6}, y' = ?

Split 5 and x^{6}.

Then log_{5} 5x^{6} = log_{5} 5 + log_{5} x^{6}.

Logarithm of a Product

log_{5} 5 = 1

log_{5} x^{6} = 6 log_{5} |x|

The number in the log cannot be minus.

So write the absolute value sign.

Logarithm of a Power

y = 1 + 6 log_{5} |x|

Then y' = 6⋅[1/(x ln 5)].

The derivative of 1 is 0.

So it's not in y'.

Arrange the right side.

So

6/(x ln 5)

is the derivative of log_{5} 5x^{6}.