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

Natural logs might seem difficult, yet once you understand a few key natural log rules, you"ll have the ability to easily settle even really complicated-looking problems. In this guide, we define the 4 most crucial natural logarithm rules, comment on other organic log properties you need to know, walk over several instances of differing difficulty, and also explain how natural logs different from other logarithms.

You are watching: E ln (x simplify)

What Is ln?

The herbal log, or ln, is the inverse of e. The letter ‘e" to represent a mathematical continuous also known as the natural exponent. Choose π, e is a mathematical constant and has a collection value. The value of e is same to roughly 2.71828.

e shows up in many instances in mathematics, consisting of scenarios about compound interest, growth equations, and also decay equations. Ln(x) is the time needed to grow to x, if ex is the quantity of expansion that has arisen after time x.

Because e is provided so commonly in math and economics, and people in these fields often have to take the logarithm v a basic of e that a number to resolve an equation or discover a value, the natural log was created as a shortcut means to write and calculate log in base e. The organic log just lets civilization reading the trouble know the you"re taking the logarithm, v a base of e, the a number. Therefore ln(x) = loge(x). Together an example, ln(5) = loge(5) = 1.609.

The 4 an essential Natural log in Rules

There are 4 main rules you need to understand when working with herbal logs, and also you"ll watch each of them again and also again in your math problems. Know these well since they deserve to be confound the first time you see them, and also you desire to make certain you have an easy rules prefer these down solid prior to moving on to more difficult logarithm topics.

Product Rule

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

Quotient Rule

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

Reciprocal Rule

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

Power Rule

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


Key organic Log Properties

In enhancement to the four natural logarithm rules discussed above, there are additionally several ln properties you need to understand if you"re studying herbal logs. have actually these memorized so girlfriend can quickly move onto the next step that the trouble without wasting time trying to remember common ln properties.


ln Property

ln of a negative Number

The ln the a negative number is undefined

ln of 0

ln(0) is undefined

ln that 1


ln the Infinity

ln(∞)= ∞

ln that e


ln of e raised to the x power

ln(ex) = x

e raised to the ln power


As you deserve to see from the final three rows, ln(e)=1, and this is true even if one is elevated to the strength of the other. This is due to the fact that the ln and e are inverse attributes of each other.

Natural log Sample Problems

Now it"s time to placed your an abilities to the test and also ensure you understand the ln rule by applying them to instance problems. Listed below are three sample problems. Try to occupational them out on her own prior to reading v the explanation.

Problem 1

Evaluate ln(72/5)

First, we usage the quotient dominance to get: ln(72) - ln(5).

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

If girlfriend don"t have actually a calculator, you can leave the equation like this, or you deserve to calculate the organic log values: 2(1.946) - 1.609 = 3.891 - 1.609 = 2.282.

Problem 2

Evaluate ln(e) /7

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

This means the trouble simplifies come 1/7, i m sorry is ours answer

Problem 3

Solve ln (5x-6)=2

When you have multiple variables within the ln parentheses, you desire to make e the base and also everything rather the exponent that e. climate you"ll acquire ln and also e next to each other and, as we recognize from the natural log rules, eln(x)=x.

So, the equation i do not care 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 number out the worth of e2, one of two people by utilizing the e vital on your calculator or using e"s approximated value of 2.718.

5x-6 =7.389

Now we"d include 6 to both sides

5x= 13.389

Finally, we"d divide both political parties by 5.

x= 2.678


How Are natural Logs various From various 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 crucial difference between natural logs and other logarithms is the basic being used. Logarithms typically use a basic of 10 (although it deserve to be a different value, which will certainly be specified), while organic logs will always use a basic of e.

This means ln(x)=loge(x)

If you should convert between logarithms and natural logs, usage the adhering to two equations:

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

Other 보다 the distinction in the basic (which is a large difference) the logarithm rules and the herbal logarithm rules room the same:

Logarithm Rules

ln Rules


ln(x)(y)= ln(x)+ln(y)



log(xa)= alog(x)

ln(xa)= aln(x)

log(10x)= x

ln(ex)= x

10log(x)= x

eln(x)= x

Summary: natural Log Rules

The herbal log, or ln, is the train station of e. The rules of herbal logs might seem counterintuitive in ~ first, however once you learn them they"re quite an easy to remember and apply to practice problems.

The 4 main ln rules are:

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

The vital difference in between natural logs and other logarithms is the base being used.

What"s Next?

Writing a research file for school yet not sure what to create about? Our overview to research document topics has over 100 subject in ten categories so you have the right to be sure to uncover the perfect topic because that you.

Want to recognize the fastest and easiest methods to convert in between Fahrenheit and Celsius? We"ve acquired you covered! check out our guide to the finest ways to convert Celsius come Fahrenheit (or vice versa).

See more: Twist My Hips Shake It Up Lyrics, Twist My Hips Lyrics

Taking the satellite or ACT? Students often struggle many with the Math section of this tests, however check the end our considerable guides to sat Math and also ACT mathematics for every little thing you require to know to ace these mathematics questions.