A Step-by-Step Guide: Finding the Derivative of log_a(x) with Respect to x

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To find the derivative of the function log_a(x) with respect to x, where “a” is the base of the logarithm:

The logarithmic function log_a(x) can be rewritten as ln(x) / ln(a), where ln(x) represents the natural logarithm of x (base e) and ln(a) represents the natural logarithm of a (base e)

To find the derivative of the function log_a(x) with respect to x, where “a” is the base of the logarithm:

The logarithmic function log_a(x) can be rewritten as ln(x) / ln(a), where ln(x) represents the natural logarithm of x (base e) and ln(a) represents the natural logarithm of a (base e).

Now, let’s find the derivative of ln(x) / ln(a) with respect to x using the quotient rule:

Step 1: Find the derivative of the numerator (ln(x)).
Using the chain rule, the derivative of ln(x) with respect to x is 1/x.

Step 2: Find the derivative of the denominator (ln(a)).
Since ln(a) is a constant (a is a constant), its derivative is 0.

Step 3: Apply the quotient rule.
Using the quotient rule, the derivative of ln(x) / ln(a) is:

[ (derivative of numerator) * ln(a) – (numerator) * (derivative of denominator) ] / (denominator)^2

Plugging in the values we found in Steps 1 and 2, we get:

[ (1/x) * ln(a) – ln(x) * 0 ] / (ln(a))^2

Simplifying further, we have:

(1/x) * ln(a) / (ln(a))^2

Now, since ln(a) is a constant, let’s express it as a single term:

(1/x) * (1 / ln(a)) / ln(a)

Finally, combining like terms, we get:

1 / (x * ln(a))

Therefore, the derivative of log_a(x) with respect to x is 1 / (x * ln(a)).

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