Derivative of log base b of x
1/ x ln(b)
The derivative of log base b of x, where b is a constant and x is a function of some variable, can be found using the following formula:
d/dx [log_b(x)] = 1 / (x * ln(b))
where ln(b) denotes the natural logarithm of b (i.e., the logarithm base e). Note that since the logarithm is only defined for positive values of x, the function x must be strictly positive.
To understand how this formula is derived, let us first recall the definition of the logarithm function. The log base b of x (written as log_b(x)) is defined as the exponent to which b must be raised to obtain x. That is,
log_b(x) = y if and only if b^y = x
Taking the derivative of both sides with respect to x yields:
d/dx [log_b(x)] = d/dx [y]
d/dx [log_b(x)] = d/dx [ln(x) / ln(b)] (since y = log_b(x) and ln(b^y) = y ln(b))
d/dx [log_b(x)] = (1 / x) * d/dx [ln(x)] – (ln(b) / x^2) * d/dx [ln(x)] (using the quotient rule for derivatives)
d/dx [log_b(x)] = [(1 / x) – (ln(b) / x^2)] * d/dx [ln(x)]
Now, we can use the fact that the derivative of ln(x) is simply 1/x, which gives us:
d/dx [log_b(x)] = 1 / (x * ln(b))
Therefore, the derivative of log base b of x is 1 divided by x multiplied by the natural logarithm of b.
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