Basic Facts About Logarithms: Rules & Examples

What Are The Basic Facts About Logarithms?

The basic facts about logarithms are the definition, the conditions on the base and argument, the special values, and the three laws that let you rewrite them. A logarithm is the inverse of raising a number to a power: it undoes exponentiation the way subtraction undoes addition. Written out, $\log_b x = y$ is just another way of saying $b^y = x$.

Everything else builds on that single equivalence. The facts split into three groups:

• The definition and its conditions: what a logarithm means, and when it is allowed to exist.

• The special values: the handful of results ($\log_b 1 = 0$, $\log_b b = 1$, and two inverse facts) that appear again and again.

• The three laws: product, quotient, and power, which convert multiplication and exponents into addition and multiplication.

Throughout this article we use a running base of $2$, so numbers stay friendly. Keep one example in mind as an anchor: $\log_2 8 = 3$, because $2^3 = 8$.

What Is A Logarithm, Exactly?

A logarithm answers one question: to what power must the base be raised to reach a given number? The number you feed in is called the argument.

$$\log_b x = y \quad \Longleftrightarrow \quad b^y = x$$

Read the left side as "log, base $b$, of $x$." The double arrow means the two statements carry the same information, written in two directions. That is why converting between them is a core skill, covered in full at log to exponential form.

Example 1: Evaluate $\log_2 8$

Ask: $2$ to what power gives $8$?

$2^3 = 8$, so the power is $3$.

Final answer: $\log_2 8 = 3$.

Because a logarithm is the inverse of an exponent, every log fact has a matching exponent fact hiding behind it. If exponents feel shaky, that is the place to firm up first.

What Conditions Must The Base And Argument Satisfy?

A logarithm is only defined when three conditions hold. Break one and the expression has no value.

• The base $b$ must be positive: $b > 0$. A negative base cannot produce a consistent set of powers.

• The base $b$ cannot equal $1$: $b \neq 1$. Since $1$ raised to any power is always $1$, no single power could ever reach a different number, so $\log_1 x$ is meaningless.

• The argument $x$ must be positive: $x > 0$. A positive base raised to any real power is always positive, so it can never equal zero or a negative number.

That last condition is the one that quietly fails students later, so hold onto it: you can never take the logarithm of $0$ or of a negative number.

What Are Common And Natural Logarithms?

Two bases show up so often that they get their own shorthand.

The common logarithm uses base $10$ and is usually written without the base at all: $\log x$ means $\log_{10} x$. It matches our base-ten number system, which is why old calculation tables, such as a printed log table, were built in base $10$.

The natural logarithm uses base $e$, where $e \approx 2.718$, and is written $\ln x$. The number $e$ arises naturally in growth, decay, and calculus, so $\ln$ is the logarithm scientists reach for most.

$$\log 1000 = 3 \quad (\text{since } 10^3 = 1000), \qquad \ln e = 1 \quad (\text{since } e^1 = e)$$

Another common base in computing is $2$, explored at log base 2, which counts how many times you can halve a number before reaching $1$.

What Are The Basic Logarithm Identities?

Four results follow straight from the definition. They are worth memorising because they collapse whole expressions in one step.

• $\log_b 1 = 0$, because $b^0 = 1$ for any valid base.

• $\log_b b = 1$, because $b^1 = b$.

• $\log_b b^x = x$, because the log and the power are inverse operations.

• $b^{\log_b x} = x$, the same inverse fact read the other way.

The last two say the same thing: applying a base-$b$ power and a base-$b$ logarithm in either order returns you to where you started. That is the whole meaning of "inverse."

What Are The Three Laws Of Logarithms?

Three laws do the heavy lifting. Each one turns a harder operation into an easier one, which is the reason logarithms were invented in the first place.

The product law: the log of a product is the sum of the logs.

$$\log_b (M \cdot N) = \log_b M + \log_b N$$

Example 2: Simplify $\log_2 (8 \cdot 4)$

$$\log_2 (8 \cdot 4) = \log_2 8 + \log_2 4 = 3 + 2 = 5$$

Check: $8 \cdot 4 = 32 = 2^5$, so the answer $5$ is correct.

The quotient law: the log of a quotient is the difference of the logs.

$$\log_b \left( \frac{M}{N} \right) = \log_b M - \log_b N$$

$$\log_2 \left( \frac{32}{4} \right) = \log_2 32 - \log_2 4 = 5 - 2 = 3$$

The power law: the log of a power pulls the exponent out to the front.

$$\log_b (M^k) = k \cdot \log_b M$$

$$\log_2 (8^2) = 2 \cdot \log_2 8 = 2 \cdot 3 = 6$$

Check: $8^2 = 64 = 2^6$. One more fact rounds out the set, the change-of-base rule, which lets a calculator that only knows $\log$ and $\ln$ handle any base:

$$\log_b x = \frac{\log_c x}{\log_c b}$$

For the full set with derivations, see logarithm rules and the consolidated properties of logarithms.

Which Logarithm Rules Are Allowed, And Which Only Look Right?

Most log mistakes are not arithmetic slips. They are applying a rule that does not exist, usually a tempting look-alike of a real one. This table settles it.

Table: Valid logarithm laws next to the invalid look-alikes students often write instead.

Read the middle column before you rewrite anything. If a step matches the right-hand column instead, it is one of the most common ways to lose marks on a logarithms question.

Why Do Logarithms Exist?

Logarithms were not invented to make homework longer. They exist because they solve two very practical problems that keep reappearing.

• They turn multiplication into addition. Before calculators, multiplying two long numbers by hand was slow and error-prone. Because $\log(M \cdot N) = \log M + \log N$, you could look up two logs, add them, and look up the answer. Centuries of astronomy, navigation, and engineering ran on this trick, and the slide rule was simply a logarithm scale you could slide.

• They compress huge ranges into readable numbers. Some quantities span billions to one: sound energy, earthquake strength, acidity. A logarithm shrinks that range so each step of $1$ means "ten times as much." That is why the numbers on those scales stay small even when the real quantities do not.

That second reason is the deep one. A logarithm is the natural language for anything that grows or shrinks by multiplying, which is why it shows up wherever nature scales by factors instead of by equal steps.

Who Invented Logarithms?

Logarithms are one of the rare tools where we know almost the exact moment they arrived, and the story is one of sheer patience.

Two more mathematicians shaped what Napier started:

• Henry Briggs (1561–1630, England) worked with Napier to recast logarithms in base $10$, the common logarithms that filled printed tables for the next three centuries.

• Leonhard Euler (1707–1783, Switzerland) tied logarithms to the number $e$ and to exponents, giving the natural logarithm the footing it has in modern mathematics.

Where Are Logarithms Used In The Real World?

The same handful of facts runs quietly under scales people read every day.

• Chemistry (the pH scale): acidity is $-\log_{10}$ of the hydrogen-ion concentration, so a drop of $1$ in pH means ten times more acidic.

• Seismology (the Richter scale): each whole number up the scale is about ten times the ground movement, which is why a magnitude $7$ dwarfs a magnitude $6$.

• Sound (decibels): loudness is measured logarithmically, so a $10$ dB rise is a tenfold jump in sound intensity.

• Computer science: the running time of efficient searches and sorts is written with $\log_2 n$, counting how many times a list can be halved.

• Finance and biology: compound growth and population models use natural logs to move between a rate and the time it takes to reach a target.

One idea, the logarithm, sits behind acidity, earthquakes, sound, algorithms, and growth. Mathematics is the shared language across fields that otherwise look unrelated.

What Are The Most Common Logarithm Mistakes?

These four errors account for most lost marks on logarithms, verified against university precalculus "common mistakes" guides and published lists of logarithm-property errors.

Splitting the log of a sum

Where it slips in:

A student sees $\log_b(M + N)$ and writes $\log_b M + \log_b N$, borrowing the product law but applying it to a sum.

Don't do this:

Do not break up a log of a sum. There is no law that simplifies $\log_b(M + N)$.

The correct way:

The product law applies only to a product: $\log_b(M \cdot N) = \log_b M + \log_b N$. If the inside is a sum, leave it as $\log_b(M + N)$.

Turning a quotient into a division of logs

Where it slips in:

A student rewrites $\log_b \left(\frac{M}{N}\right)$ as $\frac{\log_b M}{\log_b N}$, confusing the quotient law with ordinary division.

Don't do this:

Do not divide the two logs. The quotient law is a subtraction, and the order matters.

The correct way:

Use $\log_b \left(\frac{M}{N}\right) = \log_b M - \log_b N$. The dividing form, $\frac{\log_c x}{\log_c b}$, belongs to the separate change-of-base rule, not to this one.

Placing the exponent in the wrong spot

Where it slips in:

A student writes $\log_b(M^k) = (\log_b M)^k$, raising the whole log to the power instead of bringing the exponent down.

Don't do this:

Do not leave the exponent inside or raise the log to it.

The correct way:

The power law pulls the exponent to the front: $\log_b(M^k) = k \cdot \log_b M$. So $\log_2(8^2) = 2 \cdot \log_2 8 = 6$, not $(\log_2 8)^2 = 9$.

Taking the log of zero or a negative, or losing track of the base

Where it slips in:

A student solves a log equation, gets $x = -2$, and keeps it, or reads a bare $\log$ as base $e$ when it means base $10$.

Don't do this:

Do not accept an answer that makes the argument zero or negative, and do not assume the base.

The correct way:

Check every solution against $x > 0$ and discard any that fail. Read $\log$ as base $10$ and $\ln$ as base $e$ unless a base is written explicitly.

Practice Problems On Logarithms

Work each one, then check the answer that follows.

1. Evaluate $\log_3 81$.
   (Answer: $4$, since $3^4 = 81$.)

2. Write $\log_5 25 = 2$ in exponential form.
   (Answer: $5^2 = 25$.)

3. Simplify $\log_2 6 + \log_2 4$ as a single logarithm.
   (Answer: $\log_2 24$, by the product law.)

4. Simplify $\log_7 (49^3)$.
   (Answer: $6$, since $\log_7(49^3) = 3\log_7 49 = 3 \cdot 2 = 6$.)

5. Evaluate $\log_{10} 1 + \log_{10} 10$.
   (Answer: $0 + 1 = 1$.)

6. Explain why $\log_2(-4)$ has no value.
   (Answer: the argument must be positive, and no power of $2$ is negative, so it is undefined.)

Where Should You Go Next After Logarithms?

Logarithms open several doors at once, and each of these builds directly on the facts above.

1. Logarithm rules. Go deeper on the product, quotient, power, and change-of-base laws, with more worked expansions and condensations.

2. Exponential form. Practise moving between log and exponent statements, the conversion that underlies every log problem.

3. Antilog table. See how logs and antilogs were used together to multiply by hand, and how the inverse lookup works.

If your child is building these foundations, a live Bhanzu trainer teaches logarithms starting from the "why" (the scales and shortcuts the idea was built for) in the Bhanzu algebra program.