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Yes, you can draw the graph of $\ln x$. If you are already familiar with the graph of $\ln x$, this should be a simple task for you; if not, this will be a little more challenging but not too difficult. To proceed with drawing the $\ln x$ graph, a few simple steps are required.
In this complete guide, you will learn how to draw the graph of $\ln x$ as well as some interesting facts, definitions, and applications of the given function.
First, let’s go over some of the interesting steps involved in drawing the graph of $\ln x$.
How To Graph ln x
- Let $y = \ln x$.
- Check to see if this curve cuts the axes.
- Put $y = 0$, which will give us $x= 1$.
- And for $x=0$, $y$ goes negatively infinite.
- The domain is $x>0$, and $\ln x$ is an increasing function.
- $y” = -\dfrac{1}{ x^2}$, which shows that $\ln x$ is concaved downward.
- So we get the graph of $\ln x$ as follows:
What Is a Natural Logarithm?
A number’s natural logarithm is its logarithm to the base of the mathematical constant $e$, which is a transcendental and irrational number with an approximate value of $2.718$.
Generally, the natural logarithm of $x$ is written as $\ln x$, $\log_e x$. It is considered one of the most important functions in mathematics, with implementations in physics and biology.
Uses
Natural logarithms are logarithms that are used to solve growth and time problems. The fundamentals of natural logs and logarithms are logarithmic and exponential functions.
Logarithms can be used to solve equations where the unknown shows up as the exponent of another number. In exponential decay problems, logarithms are utilized to work out the decay constant, half-life, or unknown time. They are utilized to find solutions to problems incorporating compound interest and are useful in several fields of mathematics and science.
Properties of the Natural Logarithm
When solving a problem involving natural logarithms, you must keep several important properties in mind. Natural logarithms have the following properties:
The Product Rule
According to this rule, the logarithm of the multiplication of $a$ and $b$ is the sum of the logarithms of $a$ and $b$. That is, $\ln (a\cdot b)=\ln a+\ln b$.
Example
Let $a=2$ and $b=3$, then:
$\ln (2\cdot 3)=\ln 2+\ln 3$
To simplify it further, calculate $\ln 2$ and $\ln 3$, then add both answers.
Quotient Rule
The logarithm of the division of $a$ and $b$ gives us the difference between the logarithms of $a$ and $b$. That is, $\ln \left(\dfrac{a}{b}\right)=\ln a-\ln b$.
Example
Let $a=12$ and $b=31$, then:
$\ln \left(\dfrac{12}{31}\right)=\ln 12-\ln 31$
Power Rule
We get y times the logarithm of $a$ when we raise the logarithm of $a$ to the power of $b$. That is, $\ln a^b=b\ln a$.
Example
Let $a=4$ and $b=2$, then:
$\ln 4^2=2\ln 4$
Reciprocal Rule
The natural log of the reciprocal of $a$ is the opposite of the ln of $a$. That is, $\ln\left(\dfrac{1}{a}\right)=- \ln a$.
Example
Let $a=4$, then:
$\ln\left(\dfrac{1}{4}\right)=- \ln 4$
Natural vs Common Logarithms
The logarithm is the inverse function of exponentiation in mathematics. To put it another way, the logarithm is referred to as the power to which a number should be raised to obtain another number.
It is also known as the logarithm of base ten or the common logarithm. A logarithm’s general form is given as $\log_a y=x$.
The natural logarithm is denoted by $\ln$. It is also known as the logarithm of base $e$. In this case, $e$ is a number that is roughly equal to $2.718$. The natural logarithm (ln) is denoted by the symbols $\ln x$ or $\log_e x$.
How To Calculate Natural Logarithms
The natural log was determined utilizing logarithmic or log tables before the invention of computers and scientific calculators. Nevertheless, these tables continue to be used by students during exams.
Not only that but these tables can also be used to calculate or multiply large numbers. To determine a natural log by using a log table, adhere to the steps outlined below:
Step 1
Select the suitable logarithmic table by considering the base. Often, these log tables are designed for base$-10$ logarithms, also referred to as common logs. For instance, $\log_{10}(31.62)$ necessitates the use of a base$-10$ table.
Step 2
Search for the exact cell value at the intersections by not considering all decimal places.
Take into account the row that is marked with the first two digits of the given number and the column that is marked with the third digit of the given number.
Take, for example, $\log_{10}(31.62)$ and look up in the 31st row and 6th column, and the resulting cell value will be $0.4997$.
Step 3
If the given number has four or even more significant figures, employ this step to adapt the answer. Look for a small column header with the fourth digits of the given number and add it to the preceding value while remaining within the same row. For example, in $\log_{10}(31.62)$ look up in the 31st row, small column will be 2 having the cell value 2 and so $4997 + 2 = 4999$.
Step 4
In addition to this, add a decimal point, also referred to as a mantissa. So far, the solution to the preceding example is $0.4999$.
Step 5
Ultimately, using the trial and error method, work out the integer part which is also known as characteristic.
As a result, the final answer is $1.4999$.
Problems Involving the Natural Log
Let’s work out some problems involving the natural log to have a better understanding of how its properties are applied.
The problems are solved using the natural log properties and the calculation of the natural logarithm using a calculator, that is, a modern technique. For this purpose, consider some sample problems as follows:
Problem 1
Calculate $\ln\left(\dfrac{5^3}{7}\right)$.
Apply the quotient rule first to have $\ln 5^3-\ln 7$.
Now, apply the power rule on the first term to have $3\ln 5-\ln 7$.
Next, use the calculator to evaluate $\ln 5$ and $\ln 7$ as follows:
$3(1.609)-1.946=4.827-1.946=2.881$
Problem 2
Calculate $3\ln e$.
Recall that $\ln e=1$, so that the above problem has the answer as $3$ only.
Problem 3
Consider a slightly different example, $\ln(x-2)=3$. Find the value of $x$.
To find out the value of $x$, first, you need to remove the natural log from the left-hand side of the above equation. For this purpose, raise both the sides to the exponent of $e$ as follows:
$e^{\ln(x-2)}=e^3$
Next, use the fact that $e^{\ln x}=x$ to get: $x-2 =e^3$.
Now you can separate $x$ and find out its value in the following way:
$x=e^3+2$
$x=20.086+2=22.086$
Conclusion
We’ve gone over a significant amount of information in terms of how to draw the graph of $\ln x$, as well as definitions, properties, and examples of problems involving the natural logarithm.
Let’s sum up the information to have a better understanding of the natural logarithm and its graph:
- You can draw the graph of $\ln x$.
- Drawing the graph of $\ln x$ requires some important knowledge such as domain and concavity of $\ln x$.
- A natural logarithm has a few properties that make a problem easier to solve.
- The base of the natural log is $e$ and that of the common log is $10$.
The graph of $\ln x$ is easy to find and can be drawn using modern graphing calculators, so why not take some exponential decay problems to have a better understanding of natural log properties and the behavior of its graph? This will make you a pro in solving exponential equations in no time.
Images/mathematical drawings are created with GeoGebra.