Interactive Math & Science Lessons
Lesson
A logarithm is an exponent. log_b(x) is the power you have to raise b to in order to get x — so log_b(x) = y and by = x are one statement read in two directions.
bxylog_b(x) = y, then by = x.ln(by) = ln(x).ln(by) = y·ln(b), which leaves y·ln(b) = ln(x).ln(b): y = ln(x) / ln(b). That is the change-of-base rule, and it is the line printed in the panel above — at base 2 with x = 2 it reads 0.69 / 0.69 = 1.Three curves are drawn: your curve for the base you picked, plus ln(x) and log₁₀(x) as fixed references you can measure against. All three pass through the small dot at (1, 0), because any base raised to the power 0 gives 1. The larger dot sits where the argument equals the base, at (b, 1), so it slides along the curve as you drag the base. Hovering anywhere on the graph reports all three logarithms at that exact x.
b > 0, b ≠ 1, and x > 0. These are not bookkeeping conventions — they are exactly the conditions under which "b to what power gives x?" has one real answer.ln(b), and ln(1) = 0, so the value is undefined rather than merely large. You cannot sit there — type 1 into the base and it is replaced by 0.99, where the change-of-base line divides by a value near −0.01 and the answer leaps to −68.97 for the same x = 2. Below 1 that divisor stays negative and the whole curve turns over: still a logarithm, now decreasing.LOGARITHMS — WHICH BASE YOU ARE IN, AND HOW TO MOVE BETWEEN THEM
A logarithm asks one question — what power of b gives x? — but the base changes what the answer is for. Base e turns up wherever something grows at a rate proportional to itself; base 10 counts orders of magnitude; a base below 1 flips the whole curve; and any base at all can be converted into any other with one division. The last case is the one that surprises people: as the base approaches 1 the logarithm blows up.
01
What you know: The base is e ≈ 2.718. Written ln x, this is the logarithm that appears whenever a rate of change is proportional to the current amount.
Base: ln x = loge x
Worked example: ln 2 = 0.693, because e^0.693 = 2 — and that 0.693 is why "doubling time = ln 2 / growth rate"
Open this case: natural log
02
What you know: The base is 10. Each step of 1 in the answer means the input was multiplied by ten.
Base: log10 x
Worked example: log₁₀ 100 = 2, log₁₀ 1000 = 3 — a tenfold jump in x adds exactly 1
Open this case: common log
03
What you know: 0 < b < 1. Raising a fraction to higher powers makes it smaller, so the logarithm decreases instead of increasing and goes negative for x > 1.
Base: 0 < b < 1
Worked example: log₀.₅ 4 = −2, because 0.5⁻² = 4 — the same size as log₂ 4 = 2, with the sign flipped
Open this case: base < 1
04
What you know: You need a base your calculator does not have. Divide two logarithms in any base you do have: log x in base b = ln x / ln b.
Base: logb x = ln x / ln b
Worked example: log₄ 16 = ln 16 / ln 4 = 2.7726 / 1.3863 = 2, and the same division works with log₁₀ instead of ln
Open this case: change of base
05
What you know: b is close to 1, so ln b is close to zero. Since log x in base b = ln x / ln b, dividing by that tiny number makes the output enormous.
Base: b → 1 ⇒ ln b → 0
Worked example: log₁.₀₅ 50 = ln 50 / ln 1.05 = 3.912 / 0.04879 = 80.2 — it takes 80 steps of 5% growth to multiply by 50
Open this case: near b = 1
Practice
Predict the answer first, then use the controls above to find out. Reveal only after you have committed to a guess — that is what makes it practice.
Set the base to 0.5 and leave x at 2. Before you look — will the answer be positive or negative, and roughly how big?
log₀.₅(2) = −1 exactly. Change of base makes it ln(2)/ln(0.5), and ln(0.5) is negative, so the whole curve flips. It is exactly −1 because 0.5 raised to the power −1 is 2 — halving once, backwards.The larger dot always sits at (b, 1). Drag the base from 2 up to 9 and predict what the dot does — and why the answer is 1 the whole way.
log_b(b) = 1 for every base, because “what power of b gives b?” is always 1. The dot moves horizontally because its x-coordinate is the base you are dragging.Type 1 into the base. Predict what happens before pressing anything — does the curve go vertical, flat, or something else?
0.99. Step 4 of the derivation divides by ln(b), and ln(1) = 0, so base 1 is undefined rather than steep. At 0.99 the change-of-base line divides by a value near −0.01 and reads −68.97, a huge negative number, which is what “approaching undefined from one side” looks like.2¹⁰ = 1024 and 10³ = 1000, and they are close for a reason this panel is already printing. Find the reason, and then find how fast the coincidence decays.
Change of base turns any logarithm into a ratio of two others. The tool shows ln(2) = 0.693147 and log₁₀(2) = 0.30103, and log₂(10) is the reciprocal of the second: 3.32193.
So ten doublings, expressed in powers of ten, is 10 × 0.30103.
The exponent is 3.0103 rather than 3. That 0.0103 is the entire discrepancy, and raised as a power of ten it is 1.024.
The gap compounds, because the excess multiplies rather than adding. A kibibyte is 2.4% over a thousand, a mebibyte 4.9% over a million, and a gibibyte 7.4% over a billion.
The same 0.30103 is why a doubling of power is called 3 dB. The exact figure is 3.0103 dB, and the round number is a convenience with an error of one part in three hundred.
Answer
The tool prints ln(2) = 0.693147 and log₁₀(2) = 0.30103. The second is worth memorising outright, because it converts between the two bases everyone actually uses: bits and decimal digits. A number needs about 3.32 bits per decimal digit, a 2048-bit key is about 617 decimal digits, and any doubling process crosses a factor of a thousand every ten steps. Steps 4 and 5 are the warning attached: 2¹⁰ ≈ 10³ is an approximation that gets worse every time it is applied, which is why disk manufacturers and operating systems have disagreed about the size of a terabyte for thirty years — by 10%, and growing.
Learning path
