Logarithmic Differentiation

Logarithmic differentiation is differentiating functions by first taking logarithms and then determining them. When differentiating the logarithm of a function is more accessible than differentiating the function itself, we utilize logarithmic differentiation.

We use the derivatives of logarithmic functions to differentiate functions in the logarithmic differentiation technique. It is convenient for functions that elevate a value to a variable power and differentiate its logarithm rather than the function itself. The calculator inserts a function to distinguish logarithmic differentiation. 

Determining algebraically complex functions or functions for which the conventional principles of differentiation do not apply is known as logarithmic differentiation.

Properties of Logarithms

Here are the main properties of logarithms.

• The Product Rule for Logarithms

The logarithmic product rule states that the logarithm of a product is equal to the sum of logarithms. When we multiply similar bases, we may add the exponents to logs since they are exponents.

• The Quotient Rule for Logarithms

The second most useful logarithmic identity is the quotient rule. It states that the difference of logarithms is equivalent to the logarithm (log) of their quotient. 

We can derive this property in the algebraic form based on the relationship between exponents and logarithms and the quotient rule of exponents.

• The Power Rule for Logarithms

Suppose the logarithmic base has an exponent. Then according to this rule, the logarithm of the exponential quantity is equal to the product of the exponent value and the logarithmic base.

• The Change-of-Base Formula

In theory, this rule states that we can evaluate a non-standard-base log. Under this rule, we must convert it to a fraction of the type “(standard-base log of the argument) divided by (same-standard-base log of the non-standard-base)”.

The Method of Logarithmic Differentiation

Follow the following steps to find the differentiation of a logarithmic function.

• Take the function’s natural logarithm to differentiate it.
• Use the features of logarithmic functions to spread the previously gathered terms that were difficult to differentiate in the original function.
• Differentiate the equation that results.
• To get the derivative, multiply the equation by the function itself.

Logarithmic Differentiation of log x 

Logarithmic differentiation calculates a function’s derivative by first taking the logarithm and then differentiating. When the function is of the kind y = f(x)g, this technique is beneficial (x). In this sort of issue, where y is a composite function, we must first take a logarithm, converting   log (y) = g(x) log (f(x)). 

log (y) = g(x) log (f(x)). 

It presents a scenario where an exponent function is difficult. Still, we can simply differentiate it using derivatives of logarithmic functions and the chain rule after taking log on both sides of the equation.

Steps to Solve Logarithmic Differentiation Problems

The steps to discover the differentiation of logarithmic functions are as follows:

1. Take the log on both sides.
2. To eliminate the exponent, use the log attribute.
3. Now separate the variables in the equation.
4. Simplify the resulting equation.
5. Substitute the value of y back in the equation.

Some important points of Logarithmic Differentiation are as below:

• We can combine the derivative of a log with the chain rule:

ddx(ln(y))=1ydydx.

• Taking the derivative of ln(y) is sometimes easier than taking the derivative of y, and it is the only way to differentiate some functions.
• Differentiating y=f(x) with logarithmic differentiation is a straightforward technique. Take both sides’ natural logs, then distinguish both sides concerning x. You get the result after you solve for dy dx and write y in terms of x.

• For all (non-constant) functions f and g, consider the function (f(x))(g(x)). It is a function for which there is no differentiation rule; it is neither a power nor an exponential function (due to the presence of an x in the exponent and x in the base). 

Logarithmic differentiation is an excellent method for differentiating (f(x))(g(x)). Once we get the logs, we may turn off the power and utilize the product rule.

Conclusion

In mathematics, a continuous function has no breaks when plotted on a graph and is endless. Differentiability is only conceivable if the function is continuous. 

Logarithmic differentiation is preferred over simple differentiation in many areas of science and math, especially when it comes to error correction and solving exceedingly complicated functions.

Applying the quotient rule and the product in highly complicated functions is not practicable; at this point, logarithmic differentiation is preferable.