Which of the following is NOT equivalent to log 36?
You’re probably staring at a list of expressions and wondering which one is the odd one out. In practice, the trick isn’t just memorizing the answer; it’s understanding why certain forms collapse to the same value and why others don’t. Let’s unpack the math, walk through the options, and see why one of them really doesn’t belong.
What Is log 36?
When we write log 36, we’re talking about the logarithm of 36 to an implied base of 10 – the common logarithm. Simply put, it’s the exponent you’d raise 10 to in order to get 36. Even so, if you’re more comfortable with natural logs, you can think of it as ln 36 divided by ln 10. But for everyday math, just think “log 36 ≈ 1.5563.
Why It Matters / Why People Care
Knowing how to manipulate logarithms is a cornerstone of algebra, calculus, and even data science. When you can transform a log expression into an equivalent form, you’re free to:
- Simplify complex equations
- Solve for unknowns more quickly
- Compare growth rates in engineering or economics
Missing a subtle difference can lead to a wrong answer that feels mathematically plausible but is actually off by a factor of 2, 3, or more. That’s why spotting the non‑equivalent expression is a handy skill.
How It Works
Let’s break down the standard rules that let us shuffle logarithms around without changing the value Worth keeping that in mind..
1. Logarithm of a Product
log(ab) = log a + log b
This is the most common one. If you have log(36) and you know 36 = 6×6, you can write it as log 6 + log 6 It's one of those things that adds up..
2. Logarithm of a Power
log(aⁿ) = n log a
Raise a number to a power, you bring the exponent in front of the log.
3. Logarithm of a Quotient
log(a/b) = log a – log b
You can split a division into a subtraction inside the log.
4. Change‑of‑Base
log₍b₎ a = ln a / ln b
This is handy when you need a different base, but it doesn’t change the value of the log itself.
With these tools, you can test each expression against log 36. Let’s do that.
The Candidate Expressions
Let’s assume the list we’re judging looks something like this (common in practice tests):
- 2 log 6
- log 6²
- log 36
- log (6 × 6)
- log (36 ÷ 1)
Which of these is not equivalent to log 36?
1. 2 log 6
Using rule #2, we know that 2 log 6 = log(6²) = log 36. So this one is equivalent Which is the point..
2. log 6²
Same story. The exponent is inside the log, so it’s log(6²) = log 36. Equivalent again.
3. log 36
Trivial. This is the original expression. Equivalent by definition Not complicated — just consistent..
4. log (6 × 6)
Apply rule #1: log (6 × 6) = log 6 + log 6 = 2 log 6 = log 36. Equivalent Most people skip this — try not to..
5. log (36 ÷ 1)
Now, 36 ÷ 1 = 36. Logarithms are only defined for positive numbers, so log 36 is fine. The fact that we divided by 1 doesn’t change the value. So this is also equivalent Surprisingly effective..
The Odd One Out
All the expressions above are mathematically equivalent to log 36. Because of that, for example, if the list included log (6 × 6 ÷ 6) instead, that would simplify to log 6, not log 36. But if you’re looking at a typical multiple‑choice question, the trick often lies in a subtle typo or a mis‑applied rule. Or if it had log (6 × 6 × 6), that would be log 216, which is different No workaround needed..
Since the provided list doesn’t contain a clear misfit, the only way to answer the original question is to say all of them are equivalent. In real‑world problems, the non‑equivalent expression usually comes from a slip in the algebraic manipulation—like forgetting to bring the exponent outside the log, or misreading a “÷” as “×” Still holds up..
Most guides skip this. Don't.
Common Mistakes / What Most People Get Wrong
-
Forgetting the exponent rule
Thinking log 6² = (log 6)² instead of 2 log 6. That squaring inside the log changes the value entirely. -
Misreading a fraction
Seeing log (36 ÷ 1) and assuming the division does something special when it actually does nothing Simple, but easy to overlook.. -
Ignoring the base
Mixing up common logs (base 10) with natural logs (base e) can throw you off if the problem explicitly uses one base Still holds up.. -
Assuming “×” and “÷” are interchangeable
A product inside a log is additive, but a quotient is subtractive. Confusing the two leads to wrong answers Practical, not theoretical..
Practical Tips / What Actually Works
- Always write the expression in its simplest form first. If you can see 36 as 6×6, you’re already halfway there.
- Check the domain. Logarithms only accept positive numbers. If an expression could become negative or zero, it’s invalid.
- Use a calculator for a quick sanity check. Compute log 36 ≈ 1.5563 and compare your simplified expression.
- Practice with variations. Try turning log 36 into log (9×4), log (12²/4), etc., to see the rules in action.
FAQ
Q1: Does the base of the logarithm matter when comparing expressions?
A1: No. If you’re comparing two expressions that are both in the same base, they’ll be equivalent if they simplify to the same numeric value. Switching bases changes the number but not the equivalence relationship.
Q2: What if the expression has a negative number inside the log?
A2: That’s invalid. Logarithms are only defined for positive real numbers. Any expression that could produce a negative or zero inside the log is not equivalent to log 36.
Q3: Can I use the change‑of‑base formula to prove equivalence?
A3: Absolutely. If you can show that both expressions simplify to the same fraction of ln 36 and ln 10 (or whatever base you’re using), they’re equivalent.
Q4: Why does log (6 × 6) equal log 6 + log 6?
A4: That’s the product rule for logarithms. It comes from the property of exponents: 10^(log 6 + log 6) = 10^(log 6) × 10^(log 6) = 6 × 6 = 36.
Q5: Is there a quick mnemonic to remember the log rules?
A5: Think “P” for product, “Q” for quotient, “S” for power (since the exponent is “S”tarting factor). P‑Q‑S.
Closing
Spotting the non‑equivalent expression is less about memorizing a list and more about mastering the algebraic dance of logarithms. Even so, once you know the product, quotient, and power rules, you can spot the misstep in any list. And if every expression in your list turns out to be equivalent, that’s a sign you’ve got the fundamentals down—ready to tackle more complex log problems with confidence The details matter here..
