Last updated: October 2026
What log base 2 means, with a worked example
Short answer: log₂(x) is the exponent you put on 2 to get x. Since 2¹⁰ = 1024, log₂(1024) = 10. For numbers that aren't clean powers of 2, the change-of-base formula does the work: log₂(x) = ln(x) ÷ ln(2).
Worked example — log₂(1,000,000): take the natural log of the number (ln(1,000,000) ≈ 13.8155), divide by the natural log of 2 (ln(2) ≈ 0.6931), and you get ≈ 19.93. The calculator above shows this exact working for every answer, so you can follow the arithmetic instead of trusting a black box. The reverse check underneath (2^19.93 ≈ 1,000,000) proves the answer is right.
How to do log base 2 on a TI-84 or scientific calculator
Short answer: there is no log₂ button on a TI-84 — type log(x)/log(2) using the LOG key, or ln(x)/ln(2) with the LN key. Both give the identical answer.
Most calculators only ship with base-10 log and natural log buttons, which is exactly why “how to put log base 2 in calculator ti-84” is such a common search. The change-of-base formula is the universal workaround: log₂(x) = log(x) ÷ log(2), and it works with any log button your calculator has. On a TI-84, for log₂(50): press LOG, type 50)÷LOG 2), press enter — the answer is about 5.64. On phone calculators, switch to landscape for the scientific layout with the log keys.
Log base 2 table: 1 to 20
Short answer: the integer milestones are log₂(2) = 1, log₂(4) = 2, log₂(8) = 3, log₂(16) = 4. Everything between is fractional — the table below covers 1–20.
Notice the pattern: every doubling adds exactly 1 to the answer. That property — log₂(2x) = log₂(x) + 1 — is why binary logarithms measure things that grow by doubling, like memory sizes and algorithm steps.
Where log base 2 is actually used
Short answer: anywhere computers count in powers of two — bits and bytes, binary search, information theory, and music.
Bits and bytes: log₂(n) is the number of bits needed to represent n values — log₂(256) = 8, which is why a byte holds 256 values. Algorithms: binary search halves its search space each step, so searching a million sorted items takes about log₂(1,000,000) ≈ 20 comparisons. Information theory: Shannon measured information in bits, where log₂ of the possibilities is the information content. Music: an octave is a doubling of frequency, so log₂ maps pitch — 12 semitones per octave is a base-2 scale in disguise.
Frequently asked questions
What is log base 2?
Log base 2 (written log₂) asks: to what power must 2 be raised to get this number? So log₂(8) = 3 because 2³ = 8, and log₂(1024) = 10 because 2¹⁰ = 1024. It is called the binary logarithm and it is the logarithm behind computing, music intervals and information theory.
How do I put log base 2 in a TI-84 calculator?
The TI-84 has no log₂ button, so use the change-of-base formula: type log(x)/log(2) using the LOG key (base 10) — or ln(x)/ln(2) with the LN key. Both give the same answer. For example, log(1000000)/log(2) returns about 19.93. The calculator on this page shows that exact working for every answer.
What is log base 2 of 1,000,000?
About 19.93. By the change-of-base formula: log₂(1,000,000) = ln(1,000,000) ÷ ln(2) ≈ 13.8155 ÷ 0.6931 ≈ 19.93. In other words, 2¹⁹.⁹³ ≈ 1,000,000.
Why is log base 2 used in computer science?
Because computers work in binary. Log₂(n) tells you how many bits it takes to represent n values (log₂(256) = 8 bits = 1 byte), how many times you can halve n before reaching 1 (the cost of binary search), and the information content of n possibilities in bits — the foundation of Shannon's information theory.
What is the difference between log base 2 and the natural log?
Only the base: log₂ asks for powers of 2, while the natural log (ln) asks for powers of e (≈ 2.71828). They are proportional — log₂(x) = ln(x) ÷ ln(2) — so you can convert between them by multiplying or dividing by ln(2) ≈ 0.6931.
Can you take log base 2 of a negative number?
Not in the real numbers. No real power of 2 is ever negative or zero, so log₂(x) is only defined for x greater than 0. Enter a negative number or zero in the calculator above and it will tell you exactly that.
How do I calculate log base 2 of any number by hand?
Use the change-of-base formula: log₂(x) = log(x) ÷ log(2), where log can be base 10 or the natural log — any scientific calculator has both keys. Example: log₂(50) = ln(50) ÷ ln(2) ≈ 3.9120 ÷ 0.6931 ≈ 5.64.
