Log Formulas: All Logarithm Rules & Examples

What is a logarithm?

The log formula in math is a rewritten exponent. The expression $\log_b a = c$ means exactly the same thing as $b^c = a$. The base $b$ is the number being raised; the answer $c$ is the power; $a$ is the result. Every log formula listed below is a way of moving between these two forms.

In school work, two bases come up most often. Base 10 is called the common logarithm and is written $\log x$ with no subscript. Base $e$ (where $e \approx 2.71828$) is called the natural logarithm and is written $\ln x$. The rules below apply to every base.

All Log Formulas in One Table

Variable Key

Two restrictions decide whether a log expression even exists. The base $b$ must be positive and must not be 1 (a base of 1 makes the log undefined because $1^c = 1$ for every $c$). The argument must be strictly positive — there is no real-number value for $\log_b 0$ or $\log_b(-2)$.

When to Use Each Rule

The product, quotient, and power rules are tools for expanding a single complicated log into a sum of simpler logs, or condensing a sum of simple logs back into one. Most algebra problems involving logs need one of these moves. Change of base is for the moment your calculator has $\log$ and $\ln$ but you need $\log_5$ or $\log_2$. The inverse rule appears whenever you solve an equation where the variable is stuck inside a log or inside an exponent.

Worked Examples of Log Formula

Example 1: Expand $\log_2(8x^3)$ using the log rules.

Apply the product rule:

$\log_2(8x^3) = \log_2 8 + \log_2 x^3$

Apply the power rule on the second term:

$= \log_2 8 + 3\log_2 x$

Evaluate $\log_2 8$ (because $2^3 = 8$):

$= 3 + 3\log_2 x$

Final answer: $\log_2(8x^3) = 3 + 3\log_2 x$.

Example 2: Condense $2\log_5 x - \log_5 y$ into a single log.

Apply the power rule in reverse on the first term:

$2\log_5 x = \log_5 x^2$

Substitute back:

$\log_5 x^2 - \log_5 y$

Apply the quotient rule:

$= \log_5!\left(\dfrac{x^2}{y}\right)$

Final answer: $2\log_5 x - \log_5 y = \log_5!\left(\dfrac{x^2}{y}\right)$.

Example 3: Evaluate $\log_5 12$ using change of base.

A calculator gives $\log$ (base 10) and $\ln$ (base $e$), not $\log_5$. Use change of base:

$\log_5 12 = \dfrac{\log 12}{\log 5}$

Compute each:

$\log 12 \approx 1.0792$

$\log 5 \approx 0.6990$

Divide:

$\log_5 12 \approx \dfrac{1.0792}{0.6990} \approx 1.5440$

Final answer: $\log_5 12 \approx 1.5440$.

Example 4: Solve $\log_3(x+2) - \log_3 x = 2$.

Combine the left side using the quotient rule:

$\log_3!\left(\dfrac{x+2}{x}\right) = 2$

Rewrite in exponential form (this is the inverse rule in action):

$\dfrac{x+2}{x} = 3^2 = 9$

Solve the linear equation:

$x + 2 = 9x$

$2 = 8x$

$x = \dfrac{1}{4}$

Check the domain — both $x$ and $x+2$ must be positive. $x = \dfrac{1}{4}$ satisfies this.

Final answer: $x = \dfrac{1}{4}$.

Where The Log Rules Come From

Every log rule is the same statement read in two languages. In exponent language, $b^x \cdot b^y = b^{x+y}$. In log language, that exact rule reads $\log_b(MN) = \log_b M + \log_b N$. The translation works because the log function is the inverse of the exponential function — whatever exponents do, logs undo, in mirror image.

Deriving the product rule. Let $\log_b M = x$ and $\log_b N = y$.

By the definition of a logarithm: $M = b^x$ and $N = b^y$.

Multiply:

$MN = b^x \cdot b^y = b^{x+y}$

Take $\log_b$ of both sides:

$\log_b(MN) = x + y$

Substitute $x$ and $y$ back:

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

That is the product rule. The quotient rule follows the same proof with division replacing multiplication and $b^{x-y}$ replacing $b^{x+y}$. The power rule follows from $(b^x)^k = b^{kx}$.

Deriving change of base. Let $\log_b x = c$, so $b^c = x$. Take $\log_a$ of both sides:

$\log_a(b^c) = \log_a x$

Apply the power rule on the left:

$c \cdot \log_a b = \log_a x$

Solve for $c$:

$c = \dfrac{\log_a x}{\log_a b}$

Substitute back: $\log_b x = \dfrac{\log_a x}{\log_a b}$. The reciprocal rule is the special case where $a = x$.

Why Logarithms Exist at All

Logarithms were invented to turn multiplication into addition. That sounds small. It changed the speed of human science.

Before electronic calculators, every astronomer, navigator, and engineer who needed to multiply large numbers did it by hand. A single multiplication of two seven-digit numbers could take twenty minutes and was easy to get wrong. The astronomer Pierre-Simon Laplace later said that logarithms "by shortening the labours, doubled the life of the astronomer."

The shortcut works for one reason. By the product rule, $\log(MN) = \log M + \log N$. So instead of multiplying two large numbers directly, you could:

• Look up the log of each number in a printed table.

• Add the two logs (a quick paper-and-pencil step).

• Look up the answer's log in reverse — what is called the "antilog."

A multiplication that took twenty minutes became an addition that took thirty seconds. From the early 1600s until the 1970s, every long-distance ship, every census calculation, and every astronomical orbit was computed this way.

Logarithms also matter today, even though calculators do the arithmetic for us. Three reasons:

• They linearise growth. Quantities that grow exponentially — populations, bank balances, computing power — become straight lines on a log scale. That is why finance and biology graphs are often plotted on logarithmic axes.

• They measure perception. Human ears, eyes, and chemical sensors respond to ratios of stimulus, not differences. The decibel scale (sound), the Richter scale (earthquakes), the pH scale (acidity), and stellar magnitudes (brightness) are all logarithmic for this reason.

• They run information theory. Every digital file's compression, every search engine's ranking, every modern cryptographic protocol uses $\log_2$ as the unit of information.

The log rules from this article are what make all of this possible. Every one of them is a property of exponents, written in mirror.

The Mathematicians Behind Logarithms

Three names sit at the start of the log story.

John Napier (1550–1617, Scotland) was a Scottish landowner and theologian who treated mathematics as a private hobby. He spent twenty years - from his late thirties to his late fifties - building the first table of logarithms by hand, working alone in his castle near Edinburgh.

His 1614 book Mirifici Logarithmorum Canonis Descriptio introduced the word logarithm to the world. Napier never lived to see how completely his invention would reshape science. He died three years after publishing the book.

Henry Briggs (1561–1630, England) was the first professor of geometry at Gresham College in London. After reading Napier's book, Briggs travelled to Scotland to meet him. The two reportedly sat in silence for fifteen minutes when they first met, each unable to speak from admiration of the other.

Briggs reworked Napier's tables to base 10 - the common logarithm every student now uses - and published the first base-10 log tables in 1617.

Leonhard Euler (1707–1783, Switzerland) was the eighteenth-century mathematician who connected logarithms to the constant $e$ and to the exponential function as we now write it. The notation $\ln x$ for the natural logarithm and the modern definition $e^{\ln x} = x$ both come from Euler.

[MATHEMATICIANS & HISTORY CALLOUT]

Title: Twenty years of arithmetic, alone

Mathematician: John Napier

Date and place: 1594–1614, Merchiston Castle, near Edinburgh, Scotland

The story: Napier did not know if his idea would work when he started. He spent two decades computing logarithms one entry at a time, with no calculator, no collaborator, and no published precedent. There were rumours during his lifetime that he kept a black rooster as a familiar and that the locals thought he was a wizard. When his table was finally finished and printed in 1614, every astronomer in Europe wanted a copy within months.

Why it matters: A single human, working in obscurity, can build a tool that the rest of the world ends up using for three hundred years. Logarithms were that tool.

Common Mistakes With Log Formulas

1. Splitting a log of a sum. $\log_b(M + N) \neq \log_b M + \log_b N$.

Where it slips in: The product rule looks like distribution, so students apply it to addition by reflex. The product rule splits a log of a product, not a log of a sum. There is no clean rule for $\log(M+N)$.

Don't do this: $\log(3 + 4) = \log 3 + \log 4$.

The correct way: $\log(3 + 4) = \log 7$ — evaluate the sum first, then take the log if needed.

2. Pulling a coefficient out of the wrong spot.

Where it slips in: The power rule says $k \cdot \log_b M = \log_b(M^k)$ — the coefficient becomes an exponent on the argument $M$, not on the whole log. Students sometimes write $2 \log 5 = (\log 5)^2$ by mistake.

Don't do this: $2 \log 5 = (\log 5)^2$.

The correct way: $2 \log 5 = \log 5^2 = \log 25$.

3. Forgetting that the argument must be positive.

Where it slips in: After solving a log equation, students substitute their answer back without checking whether the argument is still positive. An algebraically valid answer can be a domain failure.

Don't do this: Accept $x = -3$ as a solution to $\log(x + 1) = \log(-2x - 5)$ without checking.

The correct way: For every solution, verify each $\log_b(\text{argument})$ has its argument $> 0$. Reject any solution that breaks this.

4. Confusing $\log_b(MN)$ with $(\log_b M)(\log_b N)$.

Where it slips in: The product rule turns a log of a product into a sum of logs, not a product of logs. Multiplication of two logs has no simple rule.

Don't do this: $\log_2(3 \cdot 5) = (\log_2 3)(\log_2 5)$.

The correct way: $\log_2(3 \cdot 5) = \log_2 3 + \log_2 5$.

A real-world version of the wrong-base mistake

Chemistry students computing pH — defined as $\text{pH} = -\log_{10}[\text{H}^+]$ — sometimes use $\ln$ instead of $\log$ on the calculator. The result is off by a factor of about 2.3, because $\ln 10 \approx 2.303$. A 2.3-unit pH error is the difference between drinking water (pH 7) and battery acid (pH 0–1). The same single-character base mismatch — $\log$ versus $\ln$ — has shown up in pharmaceutical dosing calculations, in audio engineering decibel conversions, and in seismograph readings. The rule that prevents it is the same one as in the third common mistake above: check the base every time.

Related and Derived Formulas