Derivative of log_a x

The derivative of log_a x, [log_a x]',
is 1/(x ln a).

The derivative of log_a |x|, [log_a |x|]',
is also 1/(x ln a).

y = log₂ (x³ - 8x) is a composite function of
y = log₂ (whole) and (whole) = 2x + 7.

So the derivative of the composite function is,
the derivative of the outer function log₂ (x³ - 8x), 1/[(x³ - 8x) ln 2]
times,
the derivative of the inner function x³ - 8x, 3x² - 8.

Arrange the expression.

Then (3x² - 8)/[(x³ - 8x) ln 2] is the derivative of log₂ (x³ - 8x).

Split 5 and x⁶.
Then log₅ 5x⁶ = log₅ 5 + log₅ x⁶.

Logarithm of a Product

log₅ 5 = 1

log₅ x⁶ = 6 log₅ |x|
The number in the log cannot be minus.
So write the absolute value sign.

Logarithm of a Power

y = 1 + 6 log₅ |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 log₅ 5x⁶.