Logarithm Calculator & Laws Studio — Learn math properties step-by-step
Resolve logarithmic equations instantly with custom bases. Deep dive into step-by-step base transitions, interactive SVG curves, and exponential proofs.
🔬 Select Calculation Parameters
Specify a custom logarithmic base and argument. Calculations solve in real time.
💡 Standard Mathematical Presets
📊 Mathematical Properties
🌿 Step-by-Step Change of Base Solver
To solve arbitrary base logarithms, standard systems convert the values to Natural Logarithms (ln base e) or Common Logarithms (log₁₀).
📈 Dynamic Logarithmic Curve Function
Graph of function y = logb(x). Hover or drag across the curve to inspect math coordinate values. Click on the curve to load that point as base.
📖 Reference Table: Standard Logarithms (1 to 100)
Quickly reference integers from 1 to 100, showing their precise natural, binary, and common logarithmic values.
| Number (X) | Natural Log (ln x) | Common Log (log₁₀ x) | Binary Log (log₂ x) |
|---|---|---|---|
| 1 | 0.000000 | 0.000000 | 0.000000 |
| 2 | 0.693147 | 0.301030 | 1.000000 |
| 3 | 1.098612 | 0.477121 | 1.584963 |
| 4 | 1.386294 | 0.602060 | 2.000000 |
| 5 | 1.609438 | 0.698970 | 2.321928 |
| 6 | 1.791759 | 0.778151 | 2.584963 |
| 7 | 1.945910 | 0.845098 | 2.807355 |
| 8 | 2.079442 | 0.903090 | 3.000000 |
| 9 | 2.197225 | 0.954243 | 3.169925 |
| 10 | 2.302585 | 1.000000 | 3.321928 |
| 11 | 2.397895 | 1.041393 | 3.459432 |
| 12 | 2.484907 | 1.079181 | 3.584963 |
| 13 | 2.564949 | 1.113943 | 3.700440 |
| 14 | 2.639057 | 1.146128 | 3.807355 |
| 15 | 2.708050 | 1.176091 | 3.906891 |
| 16 | 2.772589 | 1.204120 | 4.000000 |
| 17 | 2.833213 | 1.230449 | 4.087463 |
| 18 | 2.890372 | 1.255273 | 4.169925 |
| 19 | 2.944439 | 1.278754 | 4.247928 |
| 20 | 2.995732 | 1.301030 | 4.321928 |
| 21 | 3.044522 | 1.322219 | 4.392317 |
| 22 | 3.091042 | 1.342423 | 4.459432 |
| 23 | 3.135494 | 1.361728 | 4.523562 |
| 24 | 3.178054 | 1.380211 | 4.584963 |
| 25 | 3.218876 | 1.397940 | 4.643856 |
| 26 | 3.258097 | 1.414973 | 4.700440 |
| 27 | 3.295837 | 1.431364 | 4.754888 |
| 28 | 3.332205 | 1.447158 | 4.807355 |
| 29 | 3.367296 | 1.462398 | 4.857981 |
| 30 | 3.401197 | 1.477121 | 4.906891 |
| 31 | 3.433987 | 1.491362 | 4.954196 |
| 32 | 3.465736 | 1.505150 | 5.000000 |
| 33 | 3.496508 | 1.518514 | 5.044394 |
| 34 | 3.526361 | 1.531479 | 5.087463 |
| 35 | 3.555348 | 1.544068 | 5.129283 |
| 36 | 3.583519 | 1.556303 | 5.169925 |
| 37 | 3.610918 | 1.568202 | 5.209453 |
| 38 | 3.637586 | 1.579784 | 5.247928 |
| 39 | 3.663562 | 1.591065 | 5.285402 |
| 40 | 3.688879 | 1.602060 | 5.321928 |
| 41 | 3.713572 | 1.612784 | 5.357552 |
| 42 | 3.737670 | 1.623249 | 5.392317 |
| 43 | 3.761200 | 1.633468 | 5.426265 |
| 44 | 3.784190 | 1.643453 | 5.459432 |
| 45 | 3.806662 | 1.653213 | 5.491853 |
| 46 | 3.828641 | 1.662758 | 5.523562 |
| 47 | 3.850148 | 1.672098 | 5.554589 |
| 48 | 3.871201 | 1.681241 | 5.584963 |
| 49 | 3.891820 | 1.690196 | 5.614710 |
| 50 | 3.912023 | 1.698970 | 5.643856 |
| 51 | 3.931826 | 1.707570 | 5.672425 |
| 52 | 3.951244 | 1.716003 | 5.700440 |
| 53 | 3.970292 | 1.724276 | 5.727920 |
| 54 | 3.988984 | 1.732394 | 5.754888 |
| 55 | 4.007333 | 1.740363 | 5.781360 |
| 56 | 4.025352 | 1.748188 | 5.807355 |
| 57 | 4.043051 | 1.755875 | 5.832890 |
| 58 | 4.060443 | 1.763428 | 5.857981 |
| 59 | 4.077537 | 1.770852 | 5.882643 |
| 60 | 4.094345 | 1.778151 | 5.906891 |
| 61 | 4.110874 | 1.785330 | 5.930737 |
| 62 | 4.127134 | 1.792392 | 5.954196 |
| 63 | 4.143135 | 1.799341 | 5.977280 |
| 64 | 4.158883 | 1.806180 | 6.000000 |
| 65 | 4.174387 | 1.812913 | 6.022368 |
| 66 | 4.189655 | 1.819544 | 6.044394 |
| 67 | 4.204693 | 1.826075 | 6.066089 |
| 68 | 4.219508 | 1.832509 | 6.087463 |
| 69 | 4.234107 | 1.838849 | 6.108524 |
| 70 | 4.248495 | 1.845098 | 6.129283 |
| 71 | 4.262680 | 1.851258 | 6.149747 |
| 72 | 4.276666 | 1.857332 | 6.169925 |
| 73 | 4.290459 | 1.863323 | 6.189825 |
| 74 | 4.304065 | 1.869232 | 6.209453 |
| 75 | 4.317488 | 1.875061 | 6.228819 |
| 76 | 4.330733 | 1.880814 | 6.247928 |
| 77 | 4.343805 | 1.886491 | 6.266787 |
| 78 | 4.356709 | 1.892095 | 6.285402 |
| 79 | 4.369448 | 1.897627 | 6.303781 |
| 80 | 4.382027 | 1.903090 | 6.321928 |
| 81 | 4.394449 | 1.908485 | 6.339850 |
| 82 | 4.406719 | 1.913814 | 6.357552 |
| 83 | 4.418841 | 1.919078 | 6.375039 |
| 84 | 4.430817 | 1.924279 | 6.392317 |
| 85 | 4.442651 | 1.929419 | 6.409391 |
| 86 | 4.454347 | 1.934498 | 6.426265 |
| 87 | 4.465908 | 1.939519 | 6.442943 |
| 88 | 4.477337 | 1.944483 | 6.459432 |
| 89 | 4.488636 | 1.949390 | 6.475733 |
| 90 | 4.499810 | 1.954243 | 6.491853 |
| 91 | 4.510860 | 1.959041 | 6.507795 |
| 92 | 4.521789 | 1.963788 | 6.523562 |
| 93 | 4.532599 | 1.968483 | 6.539159 |
| 94 | 4.543295 | 1.973128 | 6.554589 |
| 95 | 4.553877 | 1.977724 | 6.569856 |
| 96 | 4.564348 | 1.982271 | 6.584963 |
| 97 | 4.574711 | 1.986772 | 6.599913 |
| 98 | 4.584967 | 1.991226 | 6.614710 |
| 99 | 4.595120 | 1.995635 | 6.629357 |
| 100 | 4.605170 | 2.000000 | 6.643856 |
Overview & Capabilities
Welcome to the Logarithm Calculator and Laws Studio. This advanced studio resolves any logarithmic equation of form log_b(x) = y. It validates base constraints, generates natural logarithm fractional steps, plots interactive functional curves, and lists full references.
How to Use
Key Features
Tips & Best Practices
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Frequently Asked Questions
Q What is a logarithm?
A logarithm is the inverse operation of exponentiation. If b^y = x, then log_b(x) = y. For example, since 2^3 = 8, the logarithm of 8 with base 2 is 3, written as log₂(8) = 3.
Q Why can the base of a logarithm not be 1 or negative?
If the base were 1, then 1^y would always equal 1, making it impossible to solve equations like log₁(5). Negative bases are excluded because raising negative bases to fractional exponents yields complex numbers, preventing a continuous real function.
Q What is the change of base formula?
The change of base formula allows you to calculate a logarithm using any standard logarithm button on a standard calculator (usually ln or log₁₀). The formula is: log_b(x) = ln(x) / ln(b) = log₁₀(x) / log₁₀(b).
Q What is the difference between natural log (ln) and common log (log)?
Common logarithms use base 10 (crucial for pH levels, decibels, and Richter earthquake scales). Natural logarithms use Euler's number "e" (≈ 2.71828), which is fundamental in calculus, physics, and financial compounding models.
Q Can you take the logarithm of a negative number or zero?
No. In the real number system, you cannot take the logarithm of a negative number or zero. As the argument approaches zero from the positive side, the logarithm goes towards negative infinity. There is no real power to which you can raise a positive base to get a negative number or zero.