Logarithm

What Is a Logarithm?

A logarithm is the exponent to which a base must be raised to produce a given number. In simpler terms, a logarithm answers the question: "What exponent do I need to put on this base to get that number?" It's like being an 'exponent detective.' Logarithms are the inverse, or opposite, operation of exponentiation. Just like subtraction undoes addition and division undoes multiplication, logarithms undo exponents.

The relationship between an exponential equation and a logarithmic equation is fundamental. If you have an exponential equation like:

$$b^y=x$$

You can express the exact same relationship in logarithmic form as:

$${\log }_b(x)=y$$

Let's break down the parts of the logarithmic form:

• b is the base. It's the number being raised to a power. The base must be a positive number and not equal to $1$.
• x is the argument. It's the number we are trying to get by raising the base to a power. The argument must be positive.
• y is the logarithm. It is the exponent itself.

Think about $2^3=8$. Here, the base is $2$, the exponent is $3$, and the result is $8$. If we want to ask this using a logarithm, we would say, "What power do we need to raise $2$ to in order to get $8$?" The answer is $3$. We write this as ${\log }_2(8)=3$. They are two different ways of stating the same fact.

What Are the Common and Natural Logarithms?

While a logarithm can have any valid number as its base, there are two bases that are so frequently used they have earned special names and notation. Understanding these will make your work with logarithms much easier, especially when using a calculator.

The Common Logarithm (Base 10)

The common logarithm uses a base of $10$. It is so common that if you see a logarithm written without a base, like $\log (x)$, you should assume the base is $10$. Your calculator's 'LOG' button calculates the common logarithm.

Why base $10$? Our entire number system is base-10, so it's a natural fit for calculations. Common logs are used extensively in science and engineering to describe quantities that span vast ranges. For example, the pH scale (measuring acidity), the Richter scale (measuring earthquake intensity), and the decibel scale (measuring sound intensity) are all based on common logarithms.

The Natural Logarithm (Base e)

The natural logarithm uses a special, irrational number called $e$ as its base. The value of $e$ is approximately $2.71828$. The natural logarithm is written as $\ln (x)$. Your calculator has a dedicated 'LN' button for it.

The number $e$ might seem strange at first, but it appears naturally in many areas of mathematics and science, especially those involving continuous growth or decay, like compound interest, population growth, and radioactive decay. It's a fundamental constant of the universe, much like $\pi$.

How Do You Evaluate Simple Logarithms?

Evaluating a logarithm without a calculator means finding the exponent. The key is to convert the logarithm back into its exponential form and solve. This process is like translating a sentence from one language to another to understand it better.

Here's a simple, reliable method:

1. Set the logarithmic expression equal to a variable, usually $y$.
2. Rewrite the equation in exponential form: ${\log }_b(x)=y$ becomes $b^y=x$.
3. Examine the exponential equation and determine the value of $y$ that makes the statement true. Often, this involves rewriting the argument as the base raised to a certain power.

What Are the Key Properties of Logarithms?

Logarithms have powerful properties that come directly from the rules of exponents. These properties allow us to simplify complex logarithmic expressions and solve logarithmic equations. Mastering them is essential for working effectively with logs.

Let's assume $b$, $M$, and $N$ are positive numbers and $b\ne 1$. Let $p$ be any real number.

1. The Product Rule

The logarithm of a product is the sum of the logarithms of its factors. This rule comes from the exponent rule $x^a\cdot x^c=x^{a+c}$.

2. The Quotient Rule

The logarithm of a quotient is the difference of the logarithms of the numerator and the denominator. This comes from the exponent rule $\frac{x^a}{x^c}=x^{a-c}$.

3. The Power Rule

The logarithm of a number raised to a power is the power multiplied by the logarithm of the number. This is one of the most useful properties for solving equations. It comes from the exponent rule $(x^a{)}^c=x^{ac}$.

Special Properties

There are also a couple of identity properties that are very helpful:

• ${\log }_b(b)=1$ because $b^1=b$.
• ${\log }_b(1)=0$ because $b^0=1$.

How Do You Use Logarithm Properties to Expand and Condense Expressions?

The properties of logarithms are like tools that let you take expressions apart (expand) or put them back together (condense). This skill is crucial for simplifying problems and solving equations in algebra and calculus.

Expanding Logarithmic Expressions

To expand an expression means to rewrite a single logarithm as a sum, difference, or multiple of simpler logarithms. The goal is to have no products, quotients, or powers inside any single logarithm.

Condensing Logarithmic Expressions

To condense an expression means to do the reverse: rewrite a sum or difference of logarithms as a single logarithm. The key is to use the properties in the opposite direction.

Let's condense the expression $2\log (a)+3\log (b)-\frac{1}{2}\log (c)$.

1. Apply the Power Rule first (in reverse): Move the coefficients back up to be exponents.

$$\log (a^2)+\log (b^3)-\log (c^{1/2})$$

2. Apply the Product and Quotient Rules: Work from left to right. Terms being added become a product inside the log, and terms being subtracted become a division.

$$\log (a^2{b}^3)-\log (c^{1/2})$$

$$\log \left(\frac{a^2{b}^3}{c^{1/2}}\right)$$

3. Simplify if possible: Remember that an exponent of $\frac{1}{2}$ is a square root.

$$\log \left(\frac{a^2{b}^3}{\sqrt{c}}\right)$$

This is the final condensed form.

How Can You Calculate a Logarithm with Any Base?

You may have noticed that your calculator only has buttons for the common log ($\log$, base $10$) and the natural log ($\ln$, base $e$). So what do you do when you need to find the value of something like ${\log }_7(50)$? You can't just type it in directly. The solution is a handy formula called the Change of Base Formula.

This formula allows you to convert a logarithm from its original base to any new base you choose. Since our calculators use base $10$ and base $e$, we typically convert to one of those.

In this formula, $c$ is your new base. For practical purposes, we can write this using the common and natural logs:

$${\log }_b(x)=\frac{\log (x)}{\log (b)}\text{or}{\log }_b(x)=\frac{\ln (x)}{\ln (b)}$$

Both versions will give you the exact same answer. A helpful way to remember it is that the original argument ($x$) goes on top, and the original base ($b$) goes on the bottom.

Let's calculate ${\log }_7(50)$ using the formula.

Using the common log (base 10):

$${\log }_7(50)=\frac{\log (50)}{\log (7)}$$

Now, use your calculator:

$$\approx \frac{1.69897}{0.84510}\approx 2.0097$$

Using the natural log (base $e$):

$${\log }_7(50)=\frac{\ln (50)}{\ln (7)}$$

Using your calculator again:

$$\approx \frac{3.91202}{1.94591}\approx 2.0097$$

As you can see, the result is the same. This formula is an essential tool for evaluating any logarithm you encounter.

What Are Some Common Mistakes with Logarithms?

Logarithms have very specific rules, and it's easy to misapply them or try to invent new ones that seem logical but are incorrect. Being aware of these common pitfalls will help you avoid them.

• Mistake 1: The Log of a Sum/Difference.
  There is no property to simplify ${\log }_b(M+N)$ or ${\log }_b(M-N)$. A very common error is to think that ${\log }_b(M+N)={\log }_b(M)+{\log }_b(N)$. This is incorrect! The sum of logs comes from the log of a product, not a sum.
• Mistake 2: Confusing the Quotient Rule and Change of Base.
  The expression $\frac{{\log }_b(M)}{{\log }_b(N)}$ is not the same as ${\log }_b(M)-{\log }_b(N)$. The first is a division of two separate log values (related to the Change of Base formula), while the second is the result of applying the Quotient Rule to ${\log }_b(\frac{M}{N})$.
• Mistake 3: Misapplying the Power Rule.
  The Power Rule states that ${\log }_b(M^p)=p\cdot {\log }_b(M)$. The exponent $p$ must be on the argument $M$ only. An expression like $({\log }_{b}M{)}^p$ cannot be simplified using this rule. The parentheses make a big difference.
• Mistake 4: Forgetting the Argument Must Be Positive.
  You cannot take the logarithm of a negative number or zero. Expressions like ${\log }_5(-25)$ and $\ln (0)$ are undefined. This is because a positive base raised to any real exponent can never produce a negative or zero result. Always check your answers to make sure you aren't taking the log of a non-positive number.

Logarithm Quick Summary

Here is a quick reference table summarizing the most important concepts and properties of logarithms.