Have you ever wondered how to deal with large numbers? You might have heard about a quantity that increases exponentially. This phrase refers to a situation that an exponential function can model. The outputs of these functions rapidly increase as their inputs increase.
Logarithmic functions are the inverse functions of exponential functions. Since logarithmic functions are slowly increasing functions, they can be helpful when trying to rescale large quantities.
Fig. 1. The logarithmic function is a slowly increasing function
Furthermore, we can use the properties of logarithms to our advantage in many problem-solving scenarios, particularly in calculus. For these reasons, it is essential to learn how to find the derivatives of logarithmic functions.
Definition for the Derivative of the Logarithmic Function
A logarithmic function \( f(x) = \log_{a}x \) computes the logarithm with base \( a \) of an \(x\)-value. The base \( a \) must be a non-negative number. Its derivative is defined as the limit of its rate of change as the change becomes very small.
Let \( f(x) = \log_{a}x \) be a logarithmic function. Its derivative is defined by the following limit,
\[ f'(x) = \lim_{\Delta x \rightarrow 0} \frac{f(x+\Delta x) - f(x)}{\Delta x}.\]
In practice, you do not find the derivative of a logarithmic function using limits. The limit is found once to obtain a formula, which then is used along with some Differentiation Rules to find the derivatives of logarithmic functions.
Formulas for the Derivatives of Logarithmic Functions
As stated before, you can find the derivative of a logarithmic function using limits, but it is not the most practical way. Instead, you can use the following formula.
The derivative of the logarithmic function is given by \[ \frac{\mathrm{d}}{\mathrm{d}x}\log_{a}{x} = \left(\frac{1}{\ln{a}}\right) \left( \frac{1}{x} \right).\]
Here is a quick example.
Find the derivative of
\[f(x)=\log_{5}{x}.\]
Answer:
Begin by noticing that the base of the logarithmic function is \( 5.\) Knowing this, you can use the formula for the derivative of a logarithmic function, that is
\[f'(x)=\left(\frac{1}{\ln{5}} \right) \left( \frac{1}{x} \right).\]
Pretty straightforward right?
Derivative of a Log Function with base e
In the particular case where the base of a logarithmic function is \( e,\) that is \( f(x) = \log_{e} x,\) the function receives a special name.
If the base of a logarithm is the number \(e,\) then it is called a Natural Logarithm. The natural logarithmic function computes the natural logarithm of a variable, and it is denoted as
\[ f(x) = \ln{x}.\]
A natural logarithm has the base \( e,\) which means that
\[\ln{e}=1.\]
With this, the formula for the derivative of a natural logarithmic function becomes simpler, that is
\[ \begin{align} \frac{\mathrm{d}}{\mathrm{d}x}\ln{x} &= \left(\frac{1}{\ln{e}}\right) \left( \frac{1}{x} \right) \\ &= \left( \frac{1}{1} \right) \left( \frac{1}{x} \right) \\ &= \frac{1}{x}. \end{align}\]
The derivative of the natural logarithmic function is given by \[ \frac{\mathrm{d}}{\mathrm{d}x}\ln{x} =\frac{1}{x}.\]
Note that by knowing this formula, along with the properties of logarithms, you can differentiate any logarithmic function. Consider the logarithmic function
\[f(x)=\log_{a}{x}.\]
The above function can be rewritten using the properties of logarithms, that is
\[ \begin{align} f(x) &= \log_{a}{x} \\[0.5em] &= \frac{\ln{x}}{\ln{a}}. \end{align}\]
Since \( \ln{a} \) is a constant, you can use the Constant Multiple Rule to factor it out when differentiating the function, so
\[ \begin{align} \frac{\mathrm{d}f}{\mathrm{d}x} &= \frac{1}{\ln{a}}\frac{\mathrm{d}}{\mathrm{d}x}\ln{x} \\[0.5em] &= \left( \frac{1}{\ln{a}}\right) \left( \frac{1}{x} \right), \end{align} \]
which is the formula introduced at the start of the previous section.
Proof of the Derivative of the Logarithmic Function
The natural logarithmic function is the inverse function of the exponential function, this means that if
\[y=\ln{x},\]
then
\[e^y=x.\]
Next, differentiate both sides of the equation, that is
\[\frac{\mathrm{d}}{\mathrm{d}x} e^y = \frac{\mathrm{d}}{\mathrm{d}x} x\]
The left-hand side of the equation is the exponential function, so you can use the formula for the derivative of the exponential function. However, since \( y \) is a function of \(x,\) you must also use the Chain Rule.
\[ e^y\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{\mathrm{d}}{\mathrm{d}x} x\]
The right-hand side can be differentiated using the Power Rule, so
\[ e^y\frac{\mathrm{d}y}{\mathrm{d}x} = 1. \]
Finally, substitute back \(e^y=x\) and isolate the derivative of the natural logarithmic function, obtaining
\[ \frac{\mathrm{d}y}{\mathrm{d}x} = \frac{1}{x}. \]
Sometimes it is worth inspecting how to find derivatives by their definition using limits. This might be a little tricky, but this gives a bunch of experience! Let's dive into it!
Recall the definition of the derivative of the natural logarithmic function through limits, which is
\[\frac{\mathrm{d}}{\mathrm{d}x} \ln{x} = \lim_{\Delta x \rightarrow 0} \frac{\ln{(x+\Delta x)}-\ln{x}}{\Delta x}.\]
You can rewrite the expression inside the limit using the Quotient Property of Logarithms and the Power Property of Logarithms, that is
\[ \frac{\mathrm{d}}{\mathrm{d}x} \ln{x} = \lim_{\Delta x \rightarrow 0} \left[ \ln{\left( \frac{x+\Delta x}{\Delta x} \right)}^{\frac{1}{\Delta x}} \right] \]
Here comes the tricky part! Multiply by \( \frac{x}{x} \) in the exponent of the function, that is
\[ \begin{align} \frac{\mathrm{d}}{\mathrm{d}x} \ln{x} &= \lim_{\Delta x \rightarrow 0} \left[ \ln{\left( \frac{x+\Delta x}{\Delta x} \right)}^{\frac{1}{\Delta x}\frac{x}{x}} \right] \\[0.75em] &= \lim_{\Delta x \rightarrow 0} \left[ \ln{\left( \frac{x+\Delta x}{\Delta x} \right)}^{\frac{x}{\Delta x}\frac{1}{x}} \right] . \end{align} \]
Now use again the Power Property of Logarithms to move \( \frac{1}{x} \) from an exponent to a coefficient. You can take it out of the limit as it does not depend on \( \Delta x.\) You also need to simplify the fraction inside the natural logarithm, so
\[ \frac{\mathrm{d}}{\mathrm{d}x} \ln{x} = \frac{1}{x} \lim_{\Delta x \rightarrow 0} \left[ \ln{\left( 1+ \frac{\Delta x}{x}\right)^{\frac{x}{\Delta x}}} \right].\]
The next step is to use the properties of limits to switch the limit and the natural logarithm. You can do this because the natural logarithm is a continuous function.
\[ \frac{\mathrm{d}}{\mathrm{d}x} \ln{x} = \frac{1}{x} \ln{\left[ \lim_{\Delta x \rightarrow 0} \left( 1+\frac{\Delta x}{x}\right)^{\frac{x}{\Delta x}}\right]}.\]
Next, make the substitution
\[ u=\frac{x}{\Delta x}.\]
Because \( x>0, \) \( u \) tends to positive infinity as \( \Delta x \) tends to zero. This will let you rewrite the limit as
\[ \frac{\mathrm{d}}{\mathrm{d}x} \ln{x} = \frac{1}{x} \ln{\left[ \lim_{u \rightarrow \infty} \left( 1+ \frac{1}{u}\right)^{u}\right]},\]
which is one of the definitions of \( e,\) the base of the natural logarithm, so
\[ \frac{\mathrm{d}}{\mathrm{d}x}\ln{x} = \frac{1}{x} \ln{e}.\]
Since \( e \) its the base of the natural logarithm, this last factor is equal to 1, finally obtaining
\[ \frac{\mathrm{d}}{\mathrm{d}x} \ln{x}= \frac{1}{x}.\]
Examples of Derivatives of Logarithmic Functions
It is now time to work on some examples. You can use differentiation rules and the properties of logarithms to your advantage!
Find the derivative of
\[ f(x) = \ln{x^2}.\]
Answer:
There are two ways of finding the derivative of the given function. By using the Chain Rule, and by using properties of logarithms.
- Using the Chain Rule.Begin by letting \( u(x) = x^2,\) and use the Chain Rule, that is\[ \frac{\mathrm{d}f}{\mathrm{d}x} = \frac{\mathrm{d}}{\mathrm{d}u}\ln{u} \frac{\mathrm{d}u}{\mathrm{d}x}.\] You can find the derivative of \( u(x) \) using the Power Rule, that is\[ \frac{\mathrm{d}u}{\mathrm{d}x}=2x,\]and you can also write the derivative of the natural logarithmic function, so\[ \begin{align} \frac{\mathrm{d}f}{\mathrm{d}x} &= \left( \frac{1}{u} \right) (2x) \\[0.5em] &= \left( \frac{1}{x^2} \right) (2x) \\[0.5em] &= \frac{2}{x}. \end{align}\]
- Using Properties of Logarithms.Rather than using the Chain Rule, you can begin by rewriting the function using the power property of logarithms, that is\[ f(x)= 2\ln{x}.\]From here, you can use the Constant Multiple Rule and differentiate the natural logarithmic function, so\[ \begin{align} \frac{\mathrm{d}f}{\mathrm{d}x} &= (2)\left( \frac{1}{x} \right) \\ &= \frac{2}{x}. \end{align} \]
Which method do you prefer? You get the same answer either way!
You can use more properties of logarithms to your advantage. Consider now an example with the product property of logarithms.
Find the derivative of
\[ g(x) = \ln{\left(xe^x \right)}. \]
Answer:
Once again, you have two options for finding the derivative of the given function. Generally, it is adviced to use the properties of logarithms whenever you can.
Begin by using the product property of logarithms to rewrite the function, that is
\[ g(x) = \ln{x} + \ln{e^x}.\]Since the natural logarithmic function is the inverse function of the exponential function, you can further rewrite the above function, so
\[ g(x) = \ln{x} + x.\]
From here, you can differentiate each term, giving you
\[ \frac{\mathrm{d}g}{\mathrm{d}x} = \frac{1}{x} + 1.\]
Sometimes the properties of logarithms will not be able to be used in the function you are working with. In these cases, just apply any relevant differentiation rule.
Find the derivative of the function
\[ h(x) = \ln{\left(\sin{x}\right)}.\]
Answer:
Here you can let \( u(x) = \sin{x} \) and use the Chain Rule, that is
\[ \frac{\mathrm{d}h}{\mathrm{d}x} = \frac{\mathrm{d}}{\mathrm{d}u} \ln{u} \frac{\mathrm{d}u}{\mathrm{d}x}.\]
The derivative of the sine function is the cosine function, so
\[ \frac{\mathrm{d}u}{\mathrm{d}x} = \cos{x}.\]
Knowing this and the derivative of the natural logarithmic function lets you write
\[ \begin{align} \frac{\mathrm{d}h}{\mathrm{d}x} &= \left( \frac{1}{u} \right) (\cos{x}) \\[0.5em] &= \frac{\cos{x}}{\sin{x}} \\[0.5em] &= \tan{x}, \end{align}\]
where you have used the trigonometric identity
\[ \frac{\cos{x}}{\sin{x}}=\tan{x}.\]
Derivatives of Logarithmic Functions - Key takeaways
- Logarithmic functions are inverse functions of exponential functions of the same base.
- The natural logarithmic function is the inverse of the exponential function with base \( e.\)
- The derivative of a logarithmic function is given by\[ \frac{\mathrm{d}}{\mathrm{d}x} \log_{a}{x} = \left(\frac{1}{\ln{a}}\right) \left(\frac{1}{x} \right).\]
- In case of the natural logarithmic function, the above formula simplifies to\[ \frac{\mathrm{d}}{\mathrm{d}x} \ln{x} = \frac{1}{x}.\]
- The derivative of the natural logarithmic function can be proved by using implicit differentiation and the differentiation rule for the exponential function.
- The derivative of the natural logarithmic function can also be proved using limits. It is important to know one of the definitions of \( e \) as a limit, that is \[ e = \lim_{u \rightarrow \infty} \left( 1+\frac{1}{u} \right)^u .\]
- Properties of logarithms like the Power Rule of Logarithms and the Product Rule of Logarithms can be used before differentiating a function in order to make it simpler.
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