Natural Logarithm

What Is the Natural Logarithm?

The natural logarithm, written as $\ln (x)$, is a special type of logarithm that uses the number $e$ as its base. At its core, any logarithm is a 'power-finder'. For example, when we see ${\log }_{10}(100)$, we are asking, 'What power do I need to raise $10$ to in order to get $100$?' The answer is $2$, because ${10}^2=100$. The natural logarithm does the exact same thing, but the base is always the special number $e$, which is approximately $2.718$.

So, when you see $\ln (x)$, you can think of it as a shorthand for ${\log }_e(x)$. It answers the question:

'$e$ to what power equals $x$?'

For instance, $\ln (e)$ is asking, '$e$ to what power equals $e$?' The answer is simply $1$. If you see $\ln (7.389)$, you're asking, '$e$ to what power is $7.389$?' A calculator would tell you the answer is $2$, because $e^2\approx 7.389$.

This relationship is the most important concept to grasp. The natural logarithm finds the exponent you need to put on $e$ to get your desired number.

Who is 'e'? Meet the Magic Number

Before we can fully understand the natural logarithm, we need to get acquainted with its base, the number $e$. Just like pi ($\pi$), $e$ is an irrational number, meaning its decimal representation goes on forever without repeating. It's one of the most important constants in all of mathematics.

$e\approx 2.718281828...$

But where does it come from? Imagine you have $\mathrm{\$}1$ in a bank account that offers a generous $100\mathrm{\%}$ annual interest rate.

• If they pay you once a year, you'll have $\mathrm{\$}2$.
• If they pay you twice a year ($50\mathrm{\%}$ each time), you'll have $\mathrm{\$}2.25$.
• If they pay you monthly, you'll have about $\mathrm{\$}2.61$.
• If they pay you daily, you'll have about $\mathrm{\$}2.71$.

What if the bank paid you interest not just every day, or every hour, but every single instant? This idea of perfect, continuous growth is what $e$ represents. If you let that compounding happen infinitely often, your initial $\mathrm{\$}1$ would grow to exactly $\mathrm{\$}e$ in one year. Because of this, $e$ is often called the 'natural' base for growth, and it appears everywhere in science to model things like population growth, radioactive decay, and chemical reactions.

What Are the Key Properties of Natural Logarithms?

Natural logarithms follow the same rules as all other logarithms. Mastering these properties is essential for simplifying expressions and solving equations. Let's assume $a>0$ and $b>0$.

1. The Product Rule: The log of a product is the sum of the logs. This rule turns multiplication inside the log into addition outside the log.
   $\ln (a\cdot b)=\ln (a)+\ln (b)$
2. The Quotient Rule: The log of a quotient is the difference of the logs. This turns division inside the log into subtraction outside the log.
   $\ln \left(\frac{a}{b}\right)=\ln (a)-\ln (b)$
3. The Power Rule: The log of a number raised to a power is the power times the log of the number. This rule lets you bring an exponent down in front.
   $\ln (a^k)=k\cdot \ln (a)$

Beyond these three main rules, there are a few special properties that come directly from the definition of $\ln (x)$ as ${\log }_e(x)$:

• Log of the Base: The log of $e$ itself is $1$. This is because $e^1=e$.
  $\ln (e)=1$
• Log of One: The log of $1$ is always $0$, for any base. This is because $e^0=1$.
  $\ln (1)=0$

These properties are your toolkit for working with natural logs.

How Do You Solve Equations with Natural Logarithms?

Solving equations with $\ln (x)$ usually involves one key strategy: isolating the logarithm term and then using its inverse operation to get rid of the 'ln'. The inverse operation of taking the natural log is raising $e$ to the power of that quantity. This process is called exponentiating.

Let's walk through some examples to see how it works.

The Inverse Relationship: e^x and ln(x)

The functions $f(x)=e^x$ and $g(x)=\ln (x)$ are inverse functions. This means they 'undo' each other, just like addition undoes subtraction, or taking a square root undoes squaring a positive number. This inverse relationship is the reason our solving strategy works, and it gives us two extremely useful identities:

The first identity says that if you take the natural log of a number and then make that result the power of $e$, you get your original number back. The second identity says that if you raise $e$ to a power and then take the natural log of the result, you get your original exponent back. This second identity is perfect for solving equations where the variable is in the exponent of $e$.

Where Are Natural Logarithms Used in the Real World?

You might be wondering if you'll ever see $\ln (x)$ outside of a math classroom. The answer is a resounding yes! Because $e$ is the base of natural growth, the natural logarithm appears in many fields of science and finance to analyze processes that change over time.

• Finance: The formula for continuously compounded interest is $A=P{e}^{rt}$, where $A$ is the final amount, $P$ is the principal, $r$ is the interest rate, and $t$ is time. If you want to find out how long it will take for your money to double, you'll need to use natural logarithms to solve for $t$.
• Population Biology: Scientists model population growth using formulas based on $e$. The natural log helps them predict how long it will take for a population of bacteria, animals, or even humans to reach a certain size.
• Radioactive Decay: The decay of radioactive elements, which is the principle behind carbon dating, follows an exponential decay model. The formula is $N(t)=N_0{e}^{-\lambda t}$. Archaeologists and geologists use natural logarithms to calculate the age of ancient fossils and rocks.
• Physics and Chemistry: Many physical laws and chemical processes, from the cooling of a hot object to the pH of a solution (though often expressed in base 10), involve exponential and logarithmic relationships.

Common Mistakes to Avoid

When first learning about natural logarithms, it's easy to misapply the properties. Here are some of the most common mistakes to watch out for.

1. The Log of a Sum/Difference: A very common error is to think that the log of a sum is the sum of the logs. This is incorrect. Remember, log properties turn multiplication into addition, not addition into addition.
   • Incorrect: $\ln (x+y)=\ln (x)+\ln (y)$
   • Correct: $\ln (x\cdot y)=\ln (x)+\ln (y)$
2. The Log of a Quotient vs. a Quotient of Logs: Students often confuse the quotient rule with simply dividing two logs. These are very different operations.
   • Incorrect: $\ln \left(\frac{x}{y}\right)=\frac{\ln (x)}{\ln (y)}$
   • Correct: $\ln \left(\frac{x}{y}\right)=\ln (x)-\ln (y)$
3. Forgetting the Domain: The input of a logarithm, called the argument, must be a positive number. You cannot take the natural logarithm of zero or any negative number. Always check your answers to make sure they don't result in taking the log of a non-positive number. For example, $\ln (x-5)$ is only defined if $x-5>0$, which means $x>5$.
4. Confusing Power Rule with Exponents: The power rule is specific. An exponent on the argument can be brought down, but an exponent on the entire log cannot.
   • Correct: $\ln (x^2)=2\ln (x)$
   • Incorrect: $(\ln (x){)}^2$ is NOT the same as $2\ln (x)$. It is simply $\ln (x)\cdot \ln (x)$.

Natural Logarithm Quick Reference Guide

Here is a quick summary of the most important concepts and properties related to the natural logarithm. Use this as a study guide!