How to Find the Derivative of the Logarithm Function with Base a Using Logarithmic Differentiation Method

Derivative of loga(x)

To find the derivative of the logarithm function with base “a”, denoted as loga(x), you can use the logarithmic differentiation method

To find the derivative of the logarithm function with base “a”, denoted as loga(x), you can use the logarithmic differentiation method.

Let’s start with the logarithmic differentiation formula:

d/dx(loga(x)) = (1/ln(a)) * (1/x)

Here, ln(a) represents the natural logarithm of the base “a”, and (1/x) represents the derivative of x with respect to x, which is just 1.

Now, we substitute these values into the formula:

d/dx(loga(x)) = (1/ln(a)) * (1/x)

Simplifying this expression, we can write:

d/dx(loga(x)) = 1 / (x * ln(a))

So, the derivative of loga(x) is 1 / (x * ln(a)).

Note: If you prefer to express the final result in a different base, such as base e (natural logarithm) or base 10, you can use the following property:

loga(x) = logb(x) / logb(a)

This means that you can rewrite loga(x) in terms of a different base, and then find the derivative using the standard rules for logarithms. For example, if you want to find the derivative of log2(x), you can rewrite it as log(x) / log(2), and then differentiate it accordingly.

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