Which Of The Following Is Not Equivalent To Log36? The Answer Will Blow Your Mind!

Which of the Following Is Not Equivalent to Log 36: A Complete Guide

If you've ever stared at a multiple choice question asking which expression isn't equivalent to log 36, you know the feeling. Your brain switches off, you start second-guessing every logarithm rule you ever learned, and suddenly even the ones you were sure about feel shaky.

Here's the thing — once you understand what log 36 actually equals and how logarithmic properties work, these problems become almost fun. Not quite, but almost.

This guide will walk you through everything you need to know: what log 36 equals, why certain transformations work (and which ones don't), and how to spot the trick in test questions. Let's dig in That's the part that actually makes a difference..

What Is Log 36, Really?

Before we can figure out what isn't equivalent to log 36, we need to establish what is. In real terms, when someone writes log 36 without specifying a base, they almost always mean base 10 in pre-algebra and early algebra contexts — the common logarithm. But here's the key: the base doesn't actually matter for determining equivalence. The properties we're about to use work the same way whether it's log base 10, natural log (ln), or any other base.

So what is log 36?

The number 36 can be broken down in a few useful ways:

• 36 = 6²
• 36 = 4 × 9
• 36 = 2² × 3²

Each of these factorizations opens up a different way to express log 36 using logarithm rules. This is where the real magic happens — and where test makers love to create confusion.

The Product Rule

The product rule for logarithms states that:

log(a × b) = log(a) + log(b)

So if we factor 36 into 4 × 9, we get:

log 36 = log(4 × 9) = log 4 + log 9

This is one common equivalent form you'll see in multiple choice options Small thing, real impact..

The Power Rule

The power rule states that:

log(a^n) = n × log(a)

Since 36 = 6², we can write:

log 36 = log(6²) = 2 × log 6

Or, using the factorization 36 = 2² × 3²:

log 36 = log(2² × 3²) = 2 × log 2 + 2 × log 3

All of these are mathematically equivalent to log 36 Which is the point..

Why This Matters (And Why Test Makers Love This Topic)

Here's the deal: logarithm equivalence questions test whether you understand the properties of logarithms, not just how to calculate them on a calculator. And honestly, most students can punch log 36 into a calculator and get an answer. What trips people up is applying the rules in reverse — taking an expression like 2log6 and recognizing it equals log 36.

This matters because:

1. It shows conceptual understanding. You can't fake your way through these problems with memorized formulas. Either you get why log(6²) = 2log6, or you don't.

2. It's foundational for higher math. Logarithm properties are essential for solving equations, simplifying expressions, and later work in calculus and beyond Still holds up..

3. It's a common test question. Whether you're taking the SAT, a placement test, or just a regular algebra exam, you'll likely see something like this.

So yeah — it matters. Understanding this now saves you headaches later.

How to Determine Equivalence (Step by Step)

Here's a practical method you can use when you encounter a "which is not equivalent" question:

Step 1: Establish the Original Value

First, figure out what log 36 equals in a form you can work with. The easiest approach is usually to rewrite 36 as a power or product:

• 36 = 6²
• 36 = 4 × 9
• 36 = 2² × 3²

Pick whichever factorization seems most useful for comparing against your answer choices.

Step 2: Test Each Option

For each option, ask yourself: "Can I manipulate this to show it equals log 36 using the product rule, power rule, or quotient rule?"

Let's walk through some examples of what you might see:

Option: log(6²)
Using the power rule in reverse: log(6²) = 2log6. But wait — that's not quite log 36 yet. Actually, let me reconsider. The power rule says log(6²) = 2log6, and 6² = 36, so log(6²) = log 36. This IS equivalent.

Option: 2log6
Using the power rule: 2log6 = log(6²) = log 36. Equivalent.

Option: log4 + log9
Using the product rule: log4 + log9 = log(4 × 9) = log 36. Equivalent Less friction, more output..

Option: log3 + log12
Using the product rule: log3 + log12 = log(3 × 12) = log 36. Equivalent Small thing, real impact..

Option: log3 + log6
Using the product rule: log3 + log6 = log(3 × 6) = log 18. NOT equivalent — this equals log 18, not log 36 Surprisingly effective..

Option: log6
This is just log 6. No amount of manipulation turns log 6 into log 36 without multiplying by something. NOT equivalent.

See how it works? You apply the rules to transform each expression and check whether it simplifies to log 36.

Step 3: Identify the Odd One Out

The option that doesn't simplify to log 36 using valid logarithm properties is your answer.

Common Mistakes (What Most People Get Wrong)

After working through hundreds of these problems, I've noticed the same mistakes popping up over and over:

Confusing Addition and Multiplication

Students sometimes see log a + log b and try to multiply the bases, or see log(a × b) and try to add. The rules go in one direction: multiply inside becomes add outside (product rule), and exponents outside become multiplication inside (power rule).

Honestly, this part trips people up more than it should Simple, but easy to overlook..

Forgetting That log 36 ≠ 36

This sounds obvious, but under test pressure, people sometimes treat log 36 as if it's equal to 36. It's not. Log 36 is the exponent you would raise 10 to (or whatever the base is) to get 36. That's a completely different number Most people skip this — try not to..

Mixing Up the Quotient Rule

The quotient rule says log(a/b) = log a - log b. Some students accidentally use addition instead of subtraction, or vice versa. Keep them straight: division → subtraction Surprisingly effective..

Trying to Combine Logs That Can't Be Combined

You can only combine logarithms when they're adding or subtracting with the same base and the same overall operation. You can't, for example, combine log 2 + log 3 into log 5. That's a common error that leads to wrong answers.

Practical Tips (What Actually Works)

Here's my honest advice after years of helping students with these problems:

1. Memorize the three rules (and understand them, not just memorize). Product rule, power rule, quotient rule. Write them on your paper the second you sit down for the test. No shame in it.

2. Factor everything. When in doubt, break numbers into their prime factors or simplest products. 36 becomes 6² or 4 × 9 or 2² × 3². Having options helps you match the answer choices And that's really what it comes down to..

3. If an option has a coefficient (like 2log6), think power rule. If an option has addition or subtraction inside a log, think product or quotient rule.

4. Check the answer choices first. Sometimes you can work backwards. If you see log4 + log9 in the options and you know 4 × 9 = 36, that's almost certainly equivalent.

5. Trust your gut on simple ones. If you see just "log 6" as an option when the question is about log 36, it's almost certainly not equivalent. These problems are designed to have one obvious wrong answer.

FAQ

Is log 36 the same as log 6 + log 6?

Yes, it is. In real terms, using the product rule: log 6 + log 6 = log(6 × 6) = log 36. This is equivalent.

What is log 36 simplified?

Log 36 can be simplified using the power rule to 2log6, or using the product rule to log 4 + log 9. It can also be written as 2log2 + 2log3.

Is 2log6 equal to log 36?

Yes. Using the power rule: 2log6 = log(6²) = log 36.

What is not equivalent to log 36?

Any expression that doesn't simplify to log 36 using valid logarithm rules is not equivalent. Take this: log 3 + log 6 equals log 18, not log 36. Similarly, just log 6 by itself is not equivalent to log 36.

Can I use a calculator to check equivalence?

You can, but it's not the most efficient method. Calculate log 36 (≈ 1.5563 in base 10), then calculate each option and compare. Even so, if they match, they're equivalent. This works, but understanding the properties is faster and shows deeper knowledge.

The Bottom Line

Here's the thing: logarithm equivalence questions aren't about being a math genius. They're about knowing three simple rules and applying them consistently. Factor your number, match it to the answer choices, and use the product, power, and quotient rules to check each option.

You'll probably want to bookmark this section.

The most common wrong answers you'll see are expressions that look like they might work but don't — like adding logs that multiply to the wrong number, or forgetting to apply the power rule correctly. Once you know what to watch for, these problems become much easier.

So the next time you see "which of the following is not equivalent to log 36," you'll know exactly what to do Easy to understand, harder to ignore..