## Derivative of loga(x)

### The derivative of the logarithm function with base ‘a’ of ‘x’ can be computed using the logarithmic differentiation technique or by applying the properties of logarithms

The derivative of the logarithm function with base ‘a’ of ‘x’ can be computed using the logarithmic differentiation technique or by applying the properties of logarithms.

Using logarithmic differentiation:

1. Let y = loga(x).

2. Take the natural logarithm (ln) of both sides: ln(y) = ln(loga(x)).

3. Apply the property of logarithms: ln(y) = ln(x) / ln(a).

4. Differentiate both sides with respect to ‘x’: (1/y) * dy/dx = 1 / (x * ln(a)).

5. Solve for dy/dx: dy/dx = (1/y) * 1 / (x * ln(a)).

To express the derivative in terms of the original function ‘y = loga(x)’, substitute y = loga(x) back into the equation:

dy/dx = (1/loga(x)) * 1 / (x * ln(a)).

Therefore, the derivative of logarithm function with base ‘a’ of ‘x’ is given by:

dy/dx = 1 / (x * ln(a) * loga(x)),

where ‘x’ is the argument of the logarithm and ‘a’ is the base of the logarithm.

This derivative formula is useful in finding the slope of the tangent line to the graph of the logarithmic function at any given point.

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