Understanding the Derivative of Logarithm Function with Base b Using Logarithmic Differentiation

Derivative of log base b of x

The derivative of the logarithm function with base b, denoted as log base b of x or logₐ(x), can be found using the logarithmic differentiation technique

The derivative of the logarithm function with base b, denoted as log base b of x or logₐ(x), can be found using the logarithmic differentiation technique. The derivative of log base b of x is given by:

(d/dx) [log base b (x)] = (1 / (x * ln(b)))

Here, ln(b) represents the natural logarithm of the base b.

To derive this result, we start by taking the natural logarithm (ln) of both sides of the equation:

ln [log base b (x)] = ln (1 / (x * ln(b)))

Now, using the log property, we can bring the exponent of the logarithm down in front:

ln [log base b (x)] = -ln(x * ln(b))

Next, we apply the chain rule of differentiation. Let y = ln [log base b (x)]:

(d/dx) [y] = (d/dx) [-ln(x * ln(b))]

Applying the chain rule, we have:

(dy/dx) = -1 / (x * ln(b)) * (d/dx) [x * ln(b)]

(d/dx) [x * ln(b)] simplifies to ln(b) using the power rule of differentiation for constant multiples.

Therefore, we have:

(dy/dx) = -1 / (x * ln(b)) * ln(b)

The ln(b) terms cancel out, resulting in:

(dy/dx) = 1 / x

This is the derivative of y with respect to x. However, we are interested in the derivative of log base b of x, so we replace y with log base b of x:

(d/dx) [log base b (x)] = 1 / x

And finally, we can write the derivative of log base b of x as:

(d/dx) [log base b (x)] = (1 / (x * ln(b)))

This formula allows us to find the derivative of any logarithm function with a given base b.

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