Mastering The Derivative Of Logarithmic Functions With Natural Logarithms And Change Of Base Formula

Derivative of log base b of x

1/ x ln(b)

The derivative of log base b of x is given by:

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

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

Alternatively, we can use the change of base formula to express log base b of x in terms of natural logarithms as:

log_b(x) = ln(x) / ln(b)

Substituting this in the derivative expression, we get:

d/dx [log_b(x)] = d/dx [(ln(x) / ln(b))]

Using the quotient rule of differentiation, we get:

d/dx [log_b(x)] = (1/ln(b)) * d/dx [ln(x)] – (ln(x)/(ln(b))^2)*d/dx[ln(b)]

Since d/dx [ln(x)] = 1/x and d/dx[ln(b)] = 0 (since b is a constant), we get:

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

Therefore, both methods yield the same derivative expression.

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