Finding the Derivative of Log Base a of x: A Step-by-Step Guide with Logarithmic Differentiation Technique

Derivative of loga(x)

To find the derivative of log base a of x, where a is a positive constant and x is a positive variable, we can use the logarithmic differentiation technique

To find the derivative of log base a of x, where a is a positive constant and x is a positive variable, we can use the logarithmic differentiation technique.

Let’s start by assuming that y = loga(x), where a is the base of the logarithm.

By definition, this means that x = a^y.

Now, let’s take the natural logarithm (ln) of both sides to simplify the differentiation process:

ln(x) = ln(a^y)

Using the logarithm property, we can bring down the exponent:

ln(x) = y * ln(a)

Now, let’s differentiate both sides with respect to x:

(1/x) * dx/dx = (d/dx)[y * ln(a)]

Simplifying, we have:

1 = (d/dx)[y * ln(a)]

Now, let’s find the derivative of y with respect to x, which is dy/dx:

dy/dx * ln(a) = 1

Finally, solving for dy/dx, we get:

dy/dx = 1/ln(a)

Therefore, the derivative of log base a of x with respect to x is 1 divided by the natural logarithm of the base a. In other words, the derivative is a constant value that does not depend on x.

Please note that the above result holds true when a is a positive constant and x is a positive variable. If a is negative or x is negative or equal to zero, this result does not hold.

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