Natural Logarithm


If you"re taking a high school or college math class, you"ll likely cover natural logs. But what are natural logs? What is ln? Why does the letter e keep showing up?

Natural logs may seem difficult, but once you understand a few key natural log rules, you"ll be able lớn easily solve even very complicated-looking problems. In this guide, we explain the four most important natural logarithm rules, discuss other natural log properties you should know, go over several examples of varying difficulty, và explain how natural logs differ from other logarithms.

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What Is ln?

The natural log, or ln, is the inverse of e. The letter ‘e" represents a mathematical constant also known as the natural exponent. Lượt thích π, e is a mathematical constant & has a mix value. The value of e is equal lớn approximately 2.71828.

e appears in many instances in mathematics, including scenarios about compound interest, growth equations, & decay equations. Ln(x) is the time needed to lớn grow khổng lồ x, while ex is the amount of growth that has occurred after time x.

Because e is used so commonly in math và economics, and people in these fields often need khổng lồ take the logarithm with a base of e of a number lớn solve an equation or find a value, the natural log was created as a shortcut way to write và calculate log base e. The natural log simply lets people reading the problem know that you"re taking the logarithm, with a base of e, of a number. So ln(x) = loge(x). As an example, ln(5) = loge(5) = 1.609.

The 4 Key Natural Log Rules

There are four main rules you need to lớn know when working with natural logs, và you"ll see each of them again and again in your math problems. Know these well because they can be confusing the first time you see them, and you want lớn make sure you have basic rules lượt thích these down solid before moving on khổng lồ more difficult logarithm topics.

Product Rule

ln(x)(y) = ln(x) + ln(y)The natural log of the multiplication of x and y is the sum of the ln of x và ln of y.Example: ln(8)(6) = ln(8) + ln(6)

Quotient Rule

ln(x/y) = ln(x) - ln(y)The natural log of the division of x và y is the difference of the ln of x and ln of y.Example: ln(7/4) = ln(7) - ln(4)

Reciprocal Rule

ln(1/x) = −ln(x)The natural log of the reciprocal of x is the opposite of the ln of x.Example: ln(⅓)= -ln(3)

Power Rule

ln(xy) = y * ln(x)The natural log of x raised to the nguồn of y is y times the ln of x.Example: ln(52) = 2 * ln(5)


Key Natural Log Properties

In addition to the four natural logarithm rules discussed above, there are also several ln properties you need to know if you"re studying natural logs. Have these memorized so you can quickly move onto the next step of the problem without wasting time trying to lớn remember common ln properties.

Scenarioln Property
ln of a Negative NumberThe ln of a negative number is undefined
ln of 0ln(0) is undefined
ln of 1ln(1)=0
ln of Infinityln(∞)= ∞
ln of eln(e)=1
ln of e raised to lớn the x powerln(ex) = x
e raised to lớn the ln powereln(x)=x

As you can see from the final three rows, ln(e)=1, & this is true even if one is raised lớn the power of the other. This is because the ln và e are inverse functions of each other.

Natural Log Sample Problems

Now it"s time to put your skills to lớn the test và ensure you understand the ln rules by applying them to lớn example problems. Below are three sample problems. Try khổng lồ work them out on your own before reading through the explanation.

Problem 1

Evaluate ln(72/5)

First, we use the quotient rule to get: ln(72) - ln(5).

Next, we use the power rule to get: 2ln(7) -ln(5).

If you don"t have a calculator, you can leave the equation lượt thích this, or you can calculate the natural log values: 2(1.946) - 1.609 = 3.891 - 1.609 = 2.282.

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Problem 2

Evaluate ln(e) /7

For this problem, we need to remember than ln(e)=1

This means the problem simplifies lớn 1/7, which is our answer

Problem 3

Solve ln (5x-6)=2

When you have multiple variables within the ln parentheses, you want lớn make e the base & everything else the exponent of e. Then you"ll get ln và e next lớn each other and, as we know from the natural log rules, eln(x)=x.

So, the equation becomes eln(5x-6)=e2

Since eln(x)=x, eln(5x-6)= 5x-6

Therefore 5x-6= e2

Since e is a constant, you can then figure out the value of e2, either by using the e key on your calculator or using e"s estimated value of 2.718.

5x-6 =7.389

Now we"d showroom 6 to both sides

5x= 13.389

Finally, we"d divide both sides by 5.

x= 2.678


How Are Natural Logs Different From Other Logarithms?

As a reminder, a logarithm is the opposite of a power. If you take the log of a number, you"re undoing the exponent. The key difference between natural logs & other logarithms is the base being used. Logarithms typically use a base of 10 (although it can be a different value, which will be specified), while natural logs will always use a base of e.

This means ln(x)=loge(x)

If you need khổng lồ convert between logarithms and natural logs, use the following two equations:

log10(x) = ln(x) / ln(10)ln(x) = log10(x) / log10(e)

Other than the difference in the base (which is a big difference) the logarithm rules và the natural logarithm rules are the same:

Logarithm Rulesln Rules
log(xy)=log(x)+log(y)ln(x)(y)= ln(x)+ln(y)
log(xa)= alog(x)ln(xa)= aln(x)
log(10x)= xln(ex)= x
10log(x)= xeln(x)= x

Summary: Natural Log Rules

The natural log, or ln, is the inverse of e. The rules of natural logs may seem counterintuitive at first, but once you learn them they"re quite simple to lớn remember and apply to practice problems.

The four main ln rules are:

ln(x)( y) = ln(x) + ln(y)ln(x/y) = ln(x) - ln(y)ln(1/x)=−ln(x)n(x y) = y*ln(x)

The key difference between natural logs và other logarithms is the base being used.

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