What does log mean on a calculator?

On a calculator, the log key gives the logarithm in base 10, called the common logarithm, so log 10000 returns 4. The ln key is the natural logarithm, whose base is e, a special number close to 2.72. For any other base, divide two logs: log₂ 128 is log 128 divided by log 2, which gives 7.

How do you solve logarithms without a calculator?

Turn the log back into a power and build the number from its base. For log₅ 625, multiply fives until you reach it: 5, 25, 125, 625. That took four fives, so log₅ 625 = 4. If the number is not an exact power, find the two powers around it: 50 lies between 32 and 64, so log₂ 50 is between 5 and 6.

At what age do students learn logarithms?

In many school systems, students learn logarithms at about age 15 to 17, once powers feel familiar. The levels here are set for age 16. They cover single logs such as log₃ 27 as well as the rule for adding two logs. Being quick with powers such as 2⁶ = 64 makes every log here faster to find.

Why is log a + log b not log (a + b)?

Each log is an exponent, and adding exponents goes with multiplying powers, never with adding them: 2³ × 2² = 2⁵. So log₂ 3 + log₂ 4 = log₂ 12, the log of the product, not log₂ 7. The plus sign joins the two logs, not the two numbers inside them, and that is the step many students misread.

Is there free logarithm practice a teacher can run with a whole class?

Yes, this game is that practice: all three levels run free in any browser on school laptops or tablets, with no student accounts or installs. On the board, write log₂ 4 + log₂ 8. Half the class adds the two logs, 2 + 3 = 5; the other half multiplies 4 × 8 and finds log₂ 32 = 5. Matching answers show the product rule at work. The schools plan, priced per student, gives the teacher a dashboard of each student’s progress.

Math practice · reimagined

Logarithms Game

This logarithm game uses bases 2, 3, 5 and 10 with numbers up to 100000, and one level adds two logs: log₂ 7 + log₂ 6 = log₂ 42. Your rival faces the same problem, and the first right answer sends your fighter in swinging.

Ages 4–99+ · no download · plays in your browser · 23 languages

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What is a logarithm?

A logarithm tells you which power of a base makes a given number. Since 10² = 100, the logarithm of 100 in base 10 is 2, written log₁₀ 100 = 2, with the base as the small number after “log”. Fold a sheet of paper in half 4 times and it has 16 layers, so log₂ 16 = 4.

Every level of Logarithm

Example: what is log₂ 32?

The small 2 is the base, so the question is how many 2s multiply together to make 32.
Halve 32 again and again until you reach 1: 16, 8, 4, 2, 1.
That took five halvings, so 32 is five 2s multiplied together, or 2⁵.
So log₂ 32 = 5, and the answer you type is 5.

Learn it step by step

A logarithm asks the reverse of a power: "what exponent do I need?"

log₂(8) asks "2 to the what makes 8?" Since 2 × 2 × 2 = 8, that's 3 — so log₂(8) = 3. And log₁₀(1000) = 3 because 10³ = 1000 (just count the zeros).

Find the power that builds the number, and type that exponent.

Tips and common mistakes

A log and a power share the same three numbers: log₃ 81 = 4 and 3⁴ = 81. The base stays the base, and the log’s answer becomes the raised exponent.
A common slip is dividing the number by the base: log₅ 25 is not 5. Count the 5s that multiply to make 25 instead: 5 × 5, so it is 2.
For log practice off-screen, call out a number such as 243 and let your child name its base and log: 243 = 3⁵, so log₃ 243 = 5. Then swap.

How to play

Most rounds show one log, such as log₂ 64 or log₅ 125, and you type its exponent on the keypad: 6 or 3. The product rule level shows log₂7 + log₂6 = log₂? instead, and you type the product, 42. Answer first and Carrot strikes, and the less you hesitate, the more it hurts.

— Quick answers —

Logarithm: the short FAQ

What does log mean on a calculator?: On a calculator, the log key gives the logarithm in base 10, called the common logarithm, so log 10000 returns 4. The ln key is the natural logarithm, whose base is e, a special number close to 2.72. For any other base, divide two logs: log₂ 128 is log 128 divided by log 2, which gives 7.
How do you solve logarithms without a calculator?: Turn the log back into a power and build the number from its base. For log₅ 625, multiply fives until you reach it: 5, 25, 125, 625. That took four fives, so log₅ 625 = 4. If the number is not an exact power, find the two powers around it: 50 lies between 32 and 64, so log₂ 50 is between 5 and 6.
At what age do students learn logarithms?: In many school systems, students learn logarithms at about age 15 to 17, once powers feel familiar. The levels here are set for age 16. They cover single logs such as log₃ 27 as well as the rule for adding two logs. Being quick with powers such as 2⁶ = 64 makes every log here faster to find.
Why is log a + log b not log (a + b)?: Each log is an exponent, and adding exponents goes with multiplying powers, never with adding them: 2³ × 2² = 2⁵. So log₂ 3 + log₂ 4 = log₂ 12, the log of the product, not log₂ 7. The plus sign joins the two logs, not the two numbers inside them, and that is the step many students misread.
Is there free logarithm practice a teacher can run with a whole class?: Yes, this game is that practice: all three levels run free in any browser on school laptops or tablets, with no student accounts or installs. On the board, write log₂ 4 + log₂ 8. Half the class adds the two logs, 2 + 3 = 5; the other half multiplies 4 × 8 and finds log₂ 32 = 5. Matching answers show the product rule at work. The schools plan, priced per student, gives the teacher a dashboard of each student’s progress.

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