Logarithmic Functions

In mathematics, a key function that serves as the inverse of an exponential function is the logarithm function. The basic form of a logarithmic function is:

\(f(x) = \log_a(x) \) or \(y = \log_a(x) \)

Here, a > 0 and a ≠ 1. Which means that, the base of a logarithmic function must be positive and not equal to 1. The exponential function of the above log form is:

\(ay = x\)

Natural logarithm (ln) and common logarithm (log) are the two types of logarithmic functions. 

For example, \(f(x) = \ln(x - 2) \) represents a natural logarithmic function, while \(g(x) = \log(x + 5) - 2 \) represents a common logarithmic function. 

The logarithmic functions help to solve exponential equations, especially when the exponent is not an integer. For instance, \(2^x = 10\) can be transformed to \(\log_2(10) = x \), and we can easily find the value of x, even if it is not a whole number. 

The formula for converting an exponential function into a logarithmic function is:

 A logarithm indicates the exponent to which base must be raised to get a value inside the log. Logarithms cannot be used for negative numbers but can be applied to positive decimals, fractions, and whole numbers.

The logarithm of zero or negative numbers cannot be calculated. Basic logarithmic function is:

\(f(x) = log (x)\) or \(y = log (x)\)  where x>0 

The domain is the set of all positive real numbers: x > 0, or (0, ∞). The output value, y, can be any real number, positive, negative, or zero. A list of y-values for various x-values is provided here:
 

• When \(x = 1, y = log (1) = 0\)
   
• When \(x = 2, y = log(2) ≈ 0.3010\) 
   
• When \(x = 0.2, y = log(0.2) ≈ -0.6990 \)
   
• When \(x = 0.01, y = log (0.01) = - 2\)

The range of every logarithmic function is always a real number, and the domain is always greater than 0.

For example, the domain and range of the logarithmic function \(f(x) = log (x + 3)\).

The argument of the function should be greater than 0 to determine the domain. 

Now, we can solve for x:

\(x + 3 > 0\)

\(x > -3\)

Thus, the domain is x > -3 or (-3, ∞).

Next, we can find the range. It can be any real number. Therefore, the range is Range = R. 

Hence, the range of f(x) = R.  

A logarithmic graph depicts how the logarithmic function varies with different values of input x. The domain is the set of positive real numbers, while the range is a set of any real numbers. Concerning the line \(y = x\), the logarithmic and exponential function graphs are symmetrical. The graph of a logarithmic function is the reflection of its corresponding exponential function across the line \(y = x\).

• y = 0 when x = 1 because \(log_a(1) = 0\) for any base a. Then the graph has an x-intercept.  
   
• \(log\ a\ (0)\) is not defined, so the graph does not have a y-intercept. 
   

Take a look at the graphs of exponential and logarithmic functions for a better understanding. 

The key properties of logarithmic graphs are listed below:

• \(a > 0\) and a 1. It explains that the base must be greater than 0, and it should not be equal to 1. 
   
• When \(a > 1\), the logarithmic graph increases, while the graph decreases when \(0 < a < 1\). 
   
• The input to the function will always have a positive value. 
   
• The range or the output of a function can be any real number. 
   

Graphing Logarithmic Functions 

Finding the points that illustrate the behavior of a function and drawing a curve through them is known as graphing logarithmic functions. Depending on the base value, the direction of the curve either increases or decreases.

The curve increases if the base is larger than 1 (base > 1), and it decreases if the base is between 0 and 1 (0 < base < 1). To graph a logarithmic function, we have to follow certain steps, they are:

Step 1: Identify the domain and range. 

Step 2: Find the vertical asymptote by setting the argument equal to 0. Remember that logarithmic graphs have a vertical asymptote but no horizontal asymptote.

Step 3: Set the argument equal to 1 by substituting a value of x. To find the x-intercept, use the property \(log_a\ (1) = 0. \)

Step 4: Set the argument equal to the base by substituting the value of x. To find another point on the graph, use the property \(log_a (a) = 1\). 

Step 5: Draw a curve by joining the two points and extending the curve towards the vertical asymptote. Let us take a look at this example to make it easier to understand. 

Graph the logarithmic function \(f(x) = log_2 (x - 1)\)

Step 1: The basic form of logarithmic function is \(f(x) = log_a (x)\). 

Where a is the base, hence, the base is 2, and since 2 > 1, the curve will increase.  

Step 2: Now, set the argument greater than 0. 

\(x - 1 > 0\)

\(x > 1\)

Thus, domain = (1, 0)

Range = R

Step 3: Find the vertical asymptote by setting the argument equal to 0. In a logarithmic function, the argument must be positive ( > 0). 

\(x - 1 = 0\)

\(x = 1 \)

Hence, the vertical asymptote is at x = 1. 

Step 4: Next, we can find the points. 

At x = 2:

\(f(2) = log_2 (2 - 1) = log_2 (1) = 0\)

So the point is (2, 0)

At x = 3: 

\( f(3) = \log_2 (3 - 1) = \log_2 (2) = 1 \)

Hence, the point is (3, 1)

Step 5: Draw the graph by joining the points, starting from near the asymptote \(x = 1\).

Here, the curve passes through the points (2, 0) and (3, 1), and it increases slowly. The red line indicates the vertical asymptote at \(x = 1\). 

What are the Properties of Logarithmic Functions?

The key properties of logarithmic functions are useful when working with exponents and solving equations involving logarithms. 

Multiplication property:

\( \log_b (a \times b) = \log_b (a) + \log_b (b) \)

When two numbers are multiplied inside a logarithm, you can separate them into a sum of two logs. 

Division property:

\( \log_b \left(\frac{a}{b}\right) = \log_b (a) - \log_b (b) \)

Dividing two numbers inside a logarithm is equivalent to subtracting the logarithm of the denominator from the numerator. 

Change of base rule:

\( \log_b (a) = \frac{\log_c (a)}{\log_c (b)} \)

You can change the base by dividing the logarithm of the number by the logarithm of the new base. 

Power property:

\( \log_b (a^x) = x \, \log_b (a) \)

To remove the exponent inside a logarithm, multiply the logarithm by that exponent. 

Logarithm of 1:

\( \log_b (1) = 0 \)

Regardless of the base, the logarithm of 1 is always 0. 

Logarithm of the base:

\(\log_b (b) = 1\)

If the base and number inside the logarithm are the same, the result is always 1.

Derivative and Integral of Logarithmic Functions

A logarithmic function’s derivative explains how the function varies with changes in its input. The reverse of the derivative is the integral of the logarithmic function, which helps to find the original function from its rate of change. The derivative formula for the common and natural logarithmic functions are:

• The derivative of the natural logarithm (ln x): \( \frac{d}{dx} [\ln x] = \frac{1}{x} \)
   
• The derivative of logarithm with base a (loga x): \( \frac{d}{dx} (\log_a x) = \frac{1}{x \ln a} \)
   

The integral formulas of logarithmic functions are: 

The integral of the natural logarithm (ln x): 

\(\int \ln x \, dx = x(\ln x - 1) + C \)

This indicates that when we integrate ln x, we get a formula that includes x, in x, and a constant C. The accumulated area under the curve of ln x is represented by the constant C. 

The integral of the common logarithm (log x):

\(\int \log_{10} x \, dx = x\left(\log_{10} x - 1\right) + C \)

We get a formula that includes x, log x, and a constant C when we integrate log x. The area under the curve of the common logarithm is represented by the constant C.

Here are some of the tips and tricks which will help the students to master logarithmic functions.
 

1. Understand the concept of logarithms first. Parents should help their children, explaining that logarithms are the opposite of exponentials.
    
2. Try to memorize the main log rules, but also understand the way they work. Use numbers first, letters later to see the patterns.
    
3. Start working out with small numbers like, 2, 3, 4, 10 to make logs intuitive. Encourage the kids to check answers by converting back to exponential form. For example, \(\log_10 100 = 2\) because \(10^2 = 100\).
    
4. Try to visualize logarithms by drawing exponential curves and show how log is the inverse. Use a graphing calculator or app to plot \(y = \log_b x\) and \(y = b^x\) at the same time.
    

5. Relate Logs to Real-Life Situations. For example, octaves in music follow logarithmic scales. Richter scale which can be used while facing earthquakes can measure magnitude logarithmically. Interest growth over time in financial calculations can use logs. These examples help children see the usage of logs beyond the textbook.

Learning the concept of logarithmic functions helps us to apply them to various real-life situations. Here are some real-world applications of logarithmic functions: 

1. Earthquakes: Logarithmic functions are used by seismologists to measure seismic waves and analyze earthquakes. They measure the magnitude of earthquakes using the Richter scale, which represents large numbers using logarithmic functions. 
    
2. Laboratories: In scientific laboratories, scientists use the pH scale, which is based on a logarithmic scale, to measure the acidity of various solutions. For example, a solution with pH 4 is ten times more acidic than a solution with pH 5. 
    
3. Finance and economics: Shareholders and investors can use the logarithmic functions to calculate their compound interest and total investment rate. For instance, they can calculate the time it takes for an investment to reach a certain amount with a fixed interest rate.
    
4. Sound and acoustics: Decibels (dB) for sound and loudness is measured using logarithms. A sound 10 times more intense than another increases by 10 dB. It helps us in understanding why logarithms are useful for very large ranges of values.
    

5. Computer Science: Algorithms and big data uses algorithms to display or to measure their efficiency. Their complexity reduces from \(n\) to \(\log n\) making computation faster.