You're staring at a calculator. Plus, the problem says log base 3 of 81. Now, your calculator only has "log" and "ln" buttons. Now what?
This exact moment has frozen more math students than I can count. But not because the concept is hard — it's not. But because nobody bothers to explain the workaround in plain English. They hand you a formula and expect you to reverse-engineer it during a timed test.
Let's fix that right now The details matter here..
What Is Changing the Log Base
Here's the thing: most calculators only compute logarithms in two bases. Here's the thing — no base 7. Base 10 (the "log" button) and base e (the "ln" button). Even so, no base 3. That's it. No base whatever-your-homework-asks-for.
But logarithms follow rules. And one of those rules lets you rewrite any log in terms of logs your calculator can do.
The change of base formula looks like this:
logₐ(b) = log(b) / log(a)
Or with natural logs:
logₐ(b) = ln(b) / ln(a)
Both work. That's why both give the exact same answer. The base of the logs on the right side doesn't matter — as long as you use the same base for numerator and denominator.
So log₃(81) becomes log(81) ÷ log(3). Plus, because 3⁴ = 81. Type either into your calculator and you get 4. Or ln(81) ÷ ln(3). The math checks out.
Why This Formula Works
If you've never seen the derivation, it's worth thirty seconds. That means aʸ = b. Here's the thing — power rule brings down the exponent: y · log(a) = log(b). Let y = logₐ(b). Day to day, divide: y = log(b) / log(a). Take log of both sides: log(aʸ) = log(b). Done.
You don't need to memorize the derivation. But knowing it exists means you're not trusting magic — you're trusting algebra Simple, but easy to overlook..
Why It Matters
You'll hit this in algebra II, precalculus, calculus, chemistry (pH calculations), biology (population growth models), computer science (algorithm analysis), and finance (compound interest with weird compounding periods).
Standardized tests love it. The SAT, ACT, AP exams, and GRE all assume you can evaluate log₅(125) without a base-5 button. If you're stuck converting by hand, you're burning time Worth knowing..
And here's what most people miss: the change of base formula isn't just a calculator trick. It's how you graph logarithmic functions with arbitrary bases. Desmos and GeoGebra handle it natively, but if you're sketching by hand or writing code, you're rewriting logₐ(x) as log(x)/log(a) every single time Practical, not theoretical..
How to Do It on Different Calculators
We're talking about where it gets practical. But the button presses? The formula is universal. Not so much.
TI-84 Plus / TI-84 Plus CE
The TI-84 is the default high school calculator in the US. Here's the workflow:
- Press the LOG button (not LN, unless you prefer natural logs)
- Type the argument — the number you're taking the log of
- Press ÷ (divide)
- Press LOG again
- Type the base
- Press ENTER
Example: log₃(81) → LOG 81 ÷ LOG 3 ENTER → displays 4
Pro tip: The TI-84 OS 2.55MP and later (basically any calculator from the last decade) has a hidden shortcut. Press MATH, scroll down to logBASE(, select it. You'll get a template: logBASE( , ). Fill in the base first, then the argument. Way cleaner. But — and this matters — some standardized tests restrict this function. The ACT allows it. The SAT does too. But some state exams don't. Know your test rules.
TI-Nspire (CX / CX II / CAS)
Nspire handles this differently. You have two options:
Option A — Template: Press the log template key (looks like log with a subscript box). It's usually on the main keypad or under the trig menu. Fill in base, tab to argument, enter That's the part that actually makes a difference..
Option B — Manual: Same division method: log(81) / log(3) works fine. The CAS version will even simplify symbolically if you type log(81, 3) — but that syntax varies by OS version.
Casio fx-9750GII / fx-9860GII / fx-CG50
Casio buried the log base function in a menu. Here's the path:
- Press SHIFT then MATH (or just MATH on newer models)
- Select logab — sometimes labeled logₐb
- Enter base, press EXE, enter argument, press EXE
On the fx-CG50 (color screen), it's under MATH → logab. The template appears on screen. Clean, visual, hard to mess up.
If your Casio doesn't have logab (older models), fall back to division: log 81 ÷ log 3 EXE.
Sharp EL-W516 / EL-W535 / WriteView Series
Sharp puts it under 2ndF → log (the key with logₐ printed above it). You'll see a template: base in the lower box, argument in the main box. Type, press =, done.
Scientific Calculators (Non-Graphing) — TI-30XS, Casio fx-300ES, etc.
Most modern scientific calculators have a logₐ or logBASE function. On the TI-30XS MultiView, it's 2nd → log (the key says logₐ above it). Look for a key with a subscript box. Template appears. Fill it in.
On the Casio fx-300ES PLUS: SHIFT → log (logₐ printed above). Same deal.
Older scientific calculators (like the classic TI-30Xa or Casio fx-260) — you're doing division. No template. log 81 ÷ log 3 = is your only path Easy to understand, harder to ignore..
Phone Calculators
iOS (iPhone): Rotate to landscape. You get a scientific view. Tap log₁₀ (it toggles to ln and logᵧ — that's your change of base). Tap logᵧ, enter base, tap =, enter argument, tap =. It's clunky but works.
Android (Google Calculator): Swipe left for scientific. Tap log — long-press or tap the arrow next to it — select log base. Enter base, then argument.
Pro tip: Don't trust phone calculators for exams. They're fine for homework, but you can't use them on the SAT, ACT, or AP tests. Practice on the device you'll actually have on test day Small thing, real impact. But it adds up..
Online Calculators (Desmos, WolframAlpha, GeoGebra)
Desmos: type log_3(81) — the underscore creates the subscript. Practically speaking, press enter. Done That's the part that actually makes a difference. And it works..
WolframAlpha: type `log base
WolframAlpha
WolframAlpha is a powerful, web‑based computational engine that can evaluate logarithms with any base in a single step. The syntax is intuitive: simply type something like
log base 3 of 81
or the more compact
log_3(81)
and press Enter. In real terms, because the engine understands natural language, you can even ask “What is the logarithm of 81 with base 3? WolframAlpha will return the exact value (in this case 4) and, if you like, a decimal approximation. ” and receive a clear, step‑by‑step explanation.
This is the bit that actually matters in practice.
Other web‑based tools
- Symbolab – Type
log_{3}(81)in the search bar. Symbolab shows the work, including the change‑of‑base conversion, and offers a numeric result. - QuickMath – Enter
log(81,3)(the second argument is the base) and the site returns the answer instantly. - GeoGebra – While primarily a geometry tool, GeoGebra’s input line also handles logarithms. Use
log(81,3)orlog_3(81); the result appears in the algebra pane.
These online calculators are excellent for quick checks, especially when you’re learning the change‑of‑base technique. They also let you experiment with different bases and arguments without needing a physical device Still holds up..
Tips for reliable online use
- Copy the exact syntax from the calculator you’ll use on the test. To give you an idea, if the exam requires
log(81)/log(3), practice typing that format in the online tool to avoid syntax surprises. - Check for step‑by‑step views. Many sites (Symbolab, WolframAlpha) let you expand the solution, which is great for reinforcing the underlying algebra.
- Save results for reference. Bookmark the page or take a screenshot so you can revisit the worked example later.
Conclusion
Whether you’re working on a classroom assignment, a homework problem, or a timed exam, the ability to compute logarithms with non‑standard bases is a fundamental skill. Modern calculators—graphing, scientific, phone, or web‑based—each provide a pathway to the answer, but the underlying principle remains the same: the change‑of‑base formula. By mastering the key sequences on your specific device (template entry, division method, or built‑in logₐ function) and verifying results with an online calculator, you build both confidence and accuracy Less friction, more output..
Remember to practice with the exact tool you’ll have on test day. The more familiar you are with its interface, the less time you’ll spend wrestling with menus and the more focus you can devote to solving the problem itself. With a solid grasp of the change‑of‑base technique and a reliable method for checking your work, you’ll be ready to tackle any logarithm challenge that comes your way And that's really what it comes down to. Nothing fancy..
