Logarithms Explained Simply

What power do you raise 2 to in order to get 8? The answer is 3, and you just worked out your first logarithm. A logarithm is nothing more than a missing exponent. Once that idea clicks, the rules, the strange scales, and the calculator button all make sense. This guide builds logarithms from zero with small numbers you can check in your head.

What Is a Logarithm?

A logarithm is the exponent you need to turn a base into a given number. The statement log₂ 8 = 3 means exactly the same thing as 2³ = 8. The two forms are mirror images of one fact.

That is why mathematicians call the logarithm the inverse of the exponential. Exponents start with the power and give you the result. Logarithms start with the result and hand you back the power.

The notation has three parts. The small subscript is the base, the number after “log” is the result, and the answer is the exponent. Read log₅ 25 aloud as “the power of 5 that makes 25,” which is 2.

How Do Logs Measure the Size of a Number?

Base-10 logs count powers of ten, so they track how many digits a number has. The value log 1,000,000 = 6 because a million is 10⁶. A six-digit number such as 123,456 has a log of about 5.09.

The rule is simple: take the whole part of the log and add 1 to get the digit count. This trick shows that 2¹⁰⁰ has 31 digits, since 100 x log 2 is about 30.1.

Why Do the Three Log Rules Work?

Each log rule is an exponent rule written in reverse. Logs are exponents, so whatever exponents do, logs copy. You never need to memorize the rules once you see where they come from.

• Product rule: log(MN) = log M + log N. Multiplying powers adds exponents, since 10² x 10³ = 10⁵. So log(100 x 1,000) = 2 + 3 = 5.
• Quotient rule: log(M/N) = log M – log N. Dividing powers subtracts exponents. So log(8/2) = 0.903 – 0.301 = 0.602, which equals log 4.
• Power rule: log(Mⁿ) = n log M. Raising a power to a power multiplies exponents. So log 8 = log 2³ = 3 x 0.301 = 0.903.

Test any rule with numbers before trusting it on an exam. The check takes ten seconds with the free logarithm calculator for any base.

What Is the Change-of-Base Formula?

The change-of-base formula rewrites any log using a base your calculator has: log_b x = log x / log b. For example, log₂ 50 = 1.699 / 0.301 = 5.64. The keystrokes and the differences between log, ln, and log2 live in our guide on log vs ln vs log2 and when to use each.

Why Do Logs Turn Multiplication Into Addition?

Logs turn multiplication into addition because multiplying numbers adds their exponents. That single fact made logarithms one of the great time-savers in science history. Adding two long numbers by hand is far faster than multiplying them.

John Napier published the first log tables in 1614. Astronomers and navigators then looked up two logs, added them, and looked up the answer. A multiplication of two seven-digit numbers shrank to a few table lookups and one sum.

In the 1620s, William Oughtred placed two log scales side by side and slid one along the other. Sliding adds lengths, and adding log lengths multiplies the numbers. That device, the slide rule, stayed in engineering pockets until electronic calculators replaced it in the 1970s.

The same idea still runs inside modern software. Programs add logs of tiny probabilities instead of multiplying them, which keeps the results from shrinking to zero.

Where Do Log Scales Show Up in Real Life?

Log scales appear wherever values span many powers of ten, such as acidity, sound, earthquakes, and starlight. On these scales each equal step means multiplying by a fixed amount, not adding one.

• pH: each unit is a 10x change in hydrogen ion concentration, so pH 4 is 1,000 times more acidic than pH 7.
• Decibels: adding 10 dB means 10 times the sound intensity, and adding 20 dB means 100 times.
• Earthquakes: one magnitude unit means 10 times the ground motion and about 32 times the energy, since 10^1.5 = 31.6. Two units mean 1,000 times the energy.
• Star brightness: 5 magnitudes equal exactly 100 times the brightness, so each step is about 2.512x. Smaller numbers mean brighter stars.

How Do You Solve an Exponential Equation With Logs?

Take the log of both sides, then use the power rule to pull the unknown exponent down. Logs are the only tool that frees a variable trapped in an exponent.

Here is a classic example. How long does $1,000 at 6 percent a year take to reach $2,000? The equation is 1.06^t = 2.

1. Take the natural log of both sides: ln(1.06^t) = ln 2.
2. Apply the power rule: t x ln 1.06 = ln 2.
3. Divide: t = 0.6931 / 0.0583 = 11.9 years.

Check the answer by plugging it back in. The value 1.06^11.9 is about 2.0005, so $1,000 grows to roughly $2,000.50. The shortcuts, such as the rule of 72, belong to our deep dive on the mathematics of doubling time.

What Mistakes Do Beginners Make With Logs?

The most common mistake is splitting the log of a sum: log(a + b) is not log a + log b. The product rule works only for multiplication inside the log, never for addition.

Check it with real numbers. The sum log 5 + log 20 equals 2, because it is log 100. But log(5 + 20) = log 25, which is about 1.398.

Three more errors show up on nearly every quiz:

• Logs of zero or negatives: no power of a positive base ever gives 0 or a negative result, so these logs are undefined.
• Dividing the logs wrongly: log a / log b is a change of base, not log(a/b). Keep the two forms separate.
• Mixing bases: log 100 = 2 in base 10, but ln 100 = 4.605. Always confirm which base a formula expects.

FAQs About Logarithms

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This article explains the concept of logarithms for beginners and uses rounded values to three or four decimal places. Use the linked calculator for exact results. Reviewed for accuracy by Prof. Dr. Khalil Mudassar, PhD. Last updated September 27, 2026.