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Logarithmic Functions

Q: Why is the log of 1 always 0?

The logarithm of 1 is always 0 for any valid base (e.g., $ \log_b(1) = 0 $). This is because any non-zero number raised to the power of 0 is equal to 1. In exponential form, this is written as $ b^0 = 1 $.

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Ever wondered how to solve for an exponent, like finding the 'x' in $2^{x} = 10$ ? Logarithmic functions are the key! They are the inverse of exponential functions and provide a powerful tool for working with numbers that grow very quickly or are incredibly small.

Logarithmic Functions — an original Algebra911 reference diagram defining logarithmic functions with its key formula and a worked example. — Logarithmic Functions: A Beginner's Guide

What Is a Logarithmic Function?

A logarithmic function is the inverse of an exponential function. In simple terms, a logarithm answers the question: what exponent do I need to raise a specific base to in order to get a certain number? This relationship is fundamental to understanding how they work.

Let's look at the formal definition. The expression $\log_{b} (x) = y$ is read as "log base $b$ of $x$ equals $y$ ". It is equivalent to the exponential equation $b^{y} = x$ . They are two different ways of stating the exact same relationship.

\log_{b} (x) = y ⟺ b^{y} = x

To break this down, let's identify the parts:

Base ( $b$ ): This is the number being raised to a power. It must be a positive number and cannot be $1$ .
Argument ( $x$ ): This is the number we are trying to get to. The argument must always be positive. You cannot take the logarithm of a negative number or zero.
Logarithm/Exponent ( $y$ ): This is the result of the logarithm, and it represents the exponent needed to raise the base $b$ to get the argument $x$ .

Think of it like a puzzle. If you see $\log_{2} (8)$ , the question it's asking is: "To what power do I need to raise $2$ to get $8$ ?" Since we know $2^{3} = 8$ , the answer is $3$ . So, $\log_{2} (8) = 3$ . Understanding this direct link between logarithms and exponents is the most important first step to mastering them.

The Two Most Important Logarithms: Common and Natural

While a logarithm can have any valid number as its base, there are two bases that are so important and frequently used that they have their own special notation. You will find dedicated buttons for them on every scientific calculator.

The Common Logarithm (Base 10)

The common logarithm uses a base of $10$ . It's incredibly useful because our number system is base-10. When you see a logarithm written without an explicit base, like $\log (100)$ , it is assumed to be the common log.

Notation: $\log (x)$ is shorthand for $\log_{10} (x)$ .

So, $\log (1000)$ is asking, "To what power do I raise $10$ to get $1000$ ?" Since $10^{3} = 1000$ , we know that $\log (1000) = 3$ . This is why you'll often see common logs used in scientific fields that deal with large ranges of numbers.

The Natural Logarithm (Base e)

The natural logarithm uses a special, irrational number as its base, called Euler's number, denoted by the letter $e$ . The value of $e$ is approximately $2.71828$ . This number might seem random, but it appears naturally in many areas of mathematics and science, especially those related to growth, decay, and calculus.

Notation: $\ln (x)$ is shorthand for $\log_{e} (x)$ .

The natural logarithm is just as important as the common log, particularly in higher-level mathematics, physics, economics, and biology. For example, calculating compound interest that is compounded continuously requires the use of $e$ . Don't be intimidated by $e$ ; for now, just think of it as another special number, like $π$ , that serves as a useful base for logarithms.

How Do You Evaluate a Logarithm Without a Calculator?

Evaluating simple logarithms is a great way to test your understanding of their relationship with exponents. The goal is to find the value of the logarithm by converting it into an exponential equation that you can solve. Here is a simple, three-step process to follow.

Set it Equal to a Variable: Take the logarithmic expression and set it equal to a variable, like $y$ . This gives you an equation to work with.
Rewrite in Exponential Form: Use the core definition $\log_{b} (x) = y ⟺ b^{y} = x$ to convert the equation from step 1 into its equivalent exponential form. This is the most crucial step.
Solve for the Variable: Use your knowledge of exponents to solve for the variable. This often involves getting the bases on both sides of the equation to be the same.

Let's walk through an example to see this process in action.

Example 1

Evaluate the logarithm $\log_{4} (64)$ .

Step 1: Set it equal to $y$ .

\log_{4} (64) = y

Step 2: Rewrite in exponential form.

Here, the base $b$ is $4$ , the argument $x$ is $64$ , and the exponent is $y$ . So, the exponential form is:

4^{y} = 64

Step 3: Solve for $y$ .

We need to figure out what power of $4$ gives us $64$ . We can test powers of 4: $4^{1} = 4$ , $4^{2} = 16$ , $4^{3} = 64$ . We found it!

4^{y} = 4^{3}

Since the bases are the same, the exponents must be equal.

y = 3

Therefore, $\log_{4} (64) = 3$ .

What Are the Key Properties of Logarithms?

Logarithms have several powerful properties that allow us to rewrite and simplify expressions and solve complex equations. These properties are derived directly from the rules of exponents. Mastering them is essential for working effectively with logs. There are three main properties you need to know.

1. The Product Rule
This rule says that the logarithm of a product of two numbers is equal to the sum of their individual logarithms. In essence, it turns multiplication inside a log into addition outside the log.
$\log_{b} (M \cdot N) = \log_{b} (M) + \log_{b} (N)$
2. The Quotient Rule
This rule says that the logarithm of a quotient (or division) of two numbers is equal to the difference of their individual logarithms. It turns division inside a log into subtraction outside the log.
$\log_{b} (\frac{M}{N}) = \log_{b} (M) - \log_{b} (N)$
3. The Power Rule
This rule says that the logarithm of a number raised to an exponent is equal to the exponent multiplied by the logarithm of the number. It allows you to bring an exponent down from inside the argument to become a coefficient in front of the log.
$\log_{b} (M^{p}) = p \cdot \log_{b} (M)$

These three properties are the building blocks for manipulating logarithmic expressions. They are used in two primary ways: expanding a single logarithm into multiple terms, or condensing multiple logarithmic terms into a single logarithm.

How Do You Use Logarithm Properties?

The properties of logarithms are like tools in a toolbox. You can use them to take complicated expressions apart (expanding) or put simple pieces together (condensing). Let's see how both processes work.

Expanding Logarithms

Expanding means taking a single logarithm with a complex argument and breaking it down into a sum or difference of simpler logarithms. The goal is to have no products, quotients, or powers inside any single log argument.

Example 2

Expand the expression $\log_{5} (\frac{x^{3}}{25})$ .

Step 1: Apply the Quotient Rule.

The main operation inside the log is division, so we start by splitting it into two logs with subtraction.

\log_{5} (x^{3}) - \log_{5} (25)

Step 2: Apply the Power Rule.

The first term, $\log_{5} (x^{3})$ , has an exponent. We can bring the $3$ down in front as a multiplier.

3 \log_{5} (x) - \log_{5} (25)

Step 3: Simplify any remaining logarithms.

The second term, $\log_{5} (25)$ , can be evaluated. We ask, " $5$ to what power is $25$ ?" Since $5^{2} = 25$ , we know $\log_{5} (25) = 2$ .

3 \log_{5} (x) - 2

The fully expanded and simplified expression is $3 \log_{5} (x) - 2$ .

Condensing Logarithms

Condensing is the reverse process. You start with a sum or difference of logarithmic terms (that have the same base) and combine them into a single logarithm using the properties in reverse.

Example 3

Condense the expression $2 \log (x) + \log (y) - 3 \log (z)$ .

Step 1: Apply the Power Rule in reverse.

Any numbers in front of the logs can be moved up to become exponents on the arguments.

\log (x^{2}) + \log (y) - \log (z^{3})

Step 2: Apply the Product and Quotient Rules from left to right.

First, combine the two terms being added using the product rule.

\log (x^{2} y) - \log (z^{3})

Next, combine the remaining two terms using the quotient rule, since they are being subtracted.

\log (\frac{x^{2} y}{z^{3}})

The fully condensed expression is $\log (\frac{x^{2} y}{z^{3}})$ .

What Is the Change-of-Base Formula?

What if you need to evaluate a logarithm like $\log_{7} (90)$ ? It's not a simple number, and your calculator likely only has buttons for the common log ( $\log$ ) and the natural log ( $\ln$ ). How can you find the answer? This is where the change-of-base formula comes in. It's a fantastic tool that lets you convert a logarithm of any base into a quotient of logarithms with a different base that you can work with.

\log_{b} (x) = \frac{\log_{c} (x)}{\log_{c} (b)}

In this formula, $c$ is your new base of choice. Since calculators use base 10 and base $e$ , we typically rewrite the formula using one of them:

Using common log (base 10): $\log_{b} (x) = \frac{\log (x)}{\log (b)}$

Using natural log (base e): $\log_{b} (x) = \frac{\ln (x)}{\ln (b)}$

Let's solve our problem of finding $\log_{7} (90)$ . Using the common log version of the formula:

\log_{7} (90) = \frac{\log (90)}{\log (7)}

Now, you can type this directly into your calculator:

$\log (90) \approx 1.9542$
$\log (7) \approx 0.8451$

\log_{7} (90) \approx \frac{1.9542}{0.8451} \approx 2.3124

This means that $7^{2.3124}$ is approximately equal to $90$ . The change-of-base formula is an essential, practical tool for getting numerical answers for any logarithm.

Key formulas for logarithmic functions by Algebra911.

Where Are Logarithms Used in the Real World?

Logarithms aren't just an abstract math concept; they are used to measure and describe many real-world phenomena, especially when dealing with quantities that have a very wide range of possible values. Logarithmic scales help make these huge numbers more manageable.

The Richter Scale (Earthquakes): This scale measures the magnitude of earthquakes. It's a base-10 logarithmic scale. This means a magnitude 7 earthquake is 10 times more powerful than a magnitude 6, and 100 times more powerful than a magnitude 5.
The pH Scale (Acidity): In chemistry, the pH scale measures how acidic or basic a substance is. It is a negative logarithm of the concentration of hydrogen ions. A small change in pH value means a large change in acidity. For example, a pH of 3 is 10 times more acidic than a pH of 4.
The Decibel Scale (Sound): This scale measures the intensity of sound. A 30-decibel (dB) sound is 10 times more intense than a 20 dB sound. This scale helps us deal with the vast range of sound pressures our ears can detect, from a whisper to a jet engine.
Finance (Compound Interest): Logarithms are used to determine how long it will take for an investment to grow to a certain amount under compound interest.
Computer Science (Algorithm Complexity): In computer programming, logarithms are used to describe the efficiency of certain algorithms, like a binary search, which can find an item in a huge, sorted list very quickly.

Common Mistakes to Avoid with Logarithms

The properties of logarithms are precise, and it's easy to misapply them. Being aware of these common errors can help you avoid them in your own work. Here is a table of frequent mistakes and the correct concepts to remember.

The Mistake (Incorrect)	The Correct Concept	Explanation
$\log_{b} (M + N) = \log_{b} (M) + \log_{b} (N)$	$\log_{b} (M \cdot N) = \log_{b} (M) + \log_{b} (N)$	The log of a sum has no special property. The Product Rule applies only to the log of a product.
$\log_{b} (M - N) = \log_{b} (M) - \log_{b} (N)$	$\log_{b} (\frac{M}{N}) = \log_{b} (M) - \log_{b} (N)$	The log of a difference has no special property. The Quotient Rule applies only to the log of a quotient.
$(\log_{b} (M))^{p} = p \cdot \log_{b} (M)$	$\log_{b} (M^{p}) = p \cdot \log_{b} (M)$	The Power Rule applies only when the argument inside the log is raised to a power, not the entire logarithm itself.
$\frac{\log_{b} (M)}{\log_{b} (N)} = \log_{b} (M) - \log_{b} (N)$	$\log_{b} (\frac{M}{N}) = \log_{b} (M) - \log_{b} (N)$	A quotient of two logs cannot be simplified this way. Don't confuse it with the Quotient Rule, which applies to a quotient inside a single log.

Logarithms: A Quick Summary

Feeling overwhelmed? Let's boil it all down to the most important points. Keep this short list handy as a reference.

Core Definition: A logarithm is an exponent. The equation $\log_{b} (x) = y$ is just another way of writing $b^{y} = x$ .
The Big Three Properties:
Product Rule: $\log_{b} (M N) = \log_{b} (M) + \log_{b} (N)$
Quotient Rule: $\log_{b} (M / N) = \log_{b} (M) - \log_{b} (N)$
Power Rule: $\log_{b} (M^{p}) = p \cdot \log_{b} (M)$
Two Special Bases:
Common Log: $\log (x)$ means $\log_{10} (x)$ .
Natural Log: $\ln (x)$ means $\log_{e} (x)$ .
Calculator Trick (Change-of-Base): To find any $\log_{b} (x)$ , use your calculator to compute $\frac{\log (x)}{\log (b)}$ or $\frac{\ln (x)}{\ln (b)}$ .
Key Restrictions: The base $b$ must be positive and not $1$ . The argument $x$ must be positive.

Frequently Asked Questions

What is the point of logarithms?

The main point of logarithms is to solve for an unknown exponent. They are the inverse operation of exponentiation, allowing us to answer questions like "2 to what power equals 32?". They also help make very large numbers more manageable through logarithmic scales.

Can you take the log of a negative number or zero?

No, you cannot. The argument of a logarithm must always be a positive number. This is because there is no real exponent you can raise a positive base to that will result in a negative number or zero.

What is the difference between 'log' and 'ln'?

The only difference is the base. The notation 'log' on its own implies the common logarithm, which has a base of 10 ( $\log_{10}$ ). The notation 'ln' stands for the natural logarithm, which has a base of Euler's number, $e$ ( $\log_{e}$ ).

Why is the log of 1 always 0?

The logarithm of 1 is always 0 for any valid base (e.g., $\log_{b} (1) = 0$ ). This is because any non-zero number raised to the power of 0 is equal to 1. In exponential form, this is written as $b^{0} = 1$ .

How are logarithms and exponents related?

They are inverse operations, meaning they "undo" each other, just like addition and subtraction or multiplication and division. The expression $\log_{b} (b^{x}) = x$ and $b^{\log_{b} (x)} = x$ shows this inverse relationship perfectly.

Can the base of a logarithm be 1?

No, the base of a logarithm cannot be 1. This is because 1 raised to any power is always 1 (e.g., $1^{2} = 1$ , $1^{5} = 1$ ). This makes it impossible to get any other number, so it doesn't create a useful function for solving equations.

What does it mean to 'expand' a logarithm?

Expanding a logarithm means using the logarithm properties (Product, Quotient, and Power Rules) to break down a single, complex logarithm into multiple, simpler logarithms that are added or subtracted.

What does it mean to 'condense' a logarithm?

Condensing a logarithm is the reverse of expanding. It means using the logarithm properties to combine a series of logarithms that are being added or subtracted into a single, compact logarithm.