Why Log 1 Equals 0 Proof Formula and Solved Examples
The concept of Value of Log 1 plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Whether you’re using log tables, calculators, or solving equations, understanding log 1 helps simplify calculations and avoid mistakes during board exams and competitive tests.
What Is Value of Log 1?
The value of log 1 is the answer you get when you apply a logarithm to the number 1 using any valid base (except 1 or negative numbers). No matter what the base is, the logarithm of 1 always comes out to 0. This is because any number raised to the power zero is always 1. You’ll find this concept applied in areas such as simplifying equations, using scientific calculators, and solving physics or chemistry formulas involving exponents and logs.
Key Formula for Value of Log 1
Here’s the standard formula: \( \log_{b}1 = 0 \) where b > 0, b ≠ 1
| Type | Expression | Value |
|---|---|---|
| Common Logarithm | log10(1) | 0 |
| Natural Logarithm | ln(1) or loge(1) | 0 |
| Any Base (b) | logb(1) | 0 |
Cross-Disciplinary Usage
Value of log 1 is not only useful in Maths but also plays an important role in Physics, Computer Science, and daily logical reasoning. For example, it helps calculate pH values, electrical decibels, and algorithms in coding. Students preparing for JEE or NEET will see its relevance in various questions.
Step-by-Step Illustration
Let’s see why log 1 = 0 even if you change the base:
- Start with the definition: \( \log_{b}1 = x \)
- Change to exponent form: \( b^x = 1 \)
- The only exponent that makes any positive b (b ≠ 1) result in 1 is 0, so \( x = 0 \)
- Therefore, \( \log_{b}1 = 0 \) for any valid base b
Speed Trick or Vedic Shortcut
Here’s a quick shortcut to never forget the value of log 1:
- Remember that anything to the power 0 is 1:
So, \( b^0 = 1 \), for positive b ≠ 1. - Thus, whenever you see log(any base)1 in problems, just write 0 instantly.
This trick will help you save time while using log tables or calculators during exams. Vedantu’s tutors often use mnemonics like “log 1 is always none — it’s zero!” to make this easy to recite during stress.
Try These Yourself
- What is log51? (Try in your notebook)
- Evaluate ln(1) without using a calculator.
- If logx1 = 0, what can x be?
- Check a scientific calculator: What does log 1 give?
Frequent Errors and Misunderstandings
- Mixing up log 1 (which is zero) with log 0 (which is undefined!)
- Assuming the answer depends on the base.
- Writing log11 = 0 — that’s NOT allowed because base 1 is not valid for logs.
- Forgetting to apply log 1 = 0 in simplification steps and making the problem harder.
Relation to Other Concepts
The idea of value of log 1 connects closely with topics such as Exponents and Powers and Log Table. When you know why log 1 = 0 and log 10 = 1 (for base 10), simplifying logarithmic expressions in algebra, coordinate geometry, or even physics and biology formulas becomes practical and quick.
Classroom Tip
A quick way to remember the value of log 1 is this: “The log is the power you put on the base to get 1. The only power that works for any base is zero.” Teachers at Vedantu love using this visual in live sessions, so students can recall it quickly in any exam!
We explored value of log 1 — its basic definition, key formula, typical errors, real-world ties, and handy tricks to lock it in memory. Keep practicing these rules with Vedantu’s logarithm questions and concept pages to deepen your confidence. Use log 1 = 0 whenever you spot it, whether on school tests or in JEE/NEET entrance problems!
Related Vedantu Resources
- Value of Log 10
- Logarithms – Concepts & Rules
- Log Table (for practice)
- Log Base 2 – Computer Applications
FAQs on What Is the Value of Log 1 in Mathematics
1. What is the value of log 1?
The value of log 1 to any valid base is 0. This is because for any base a (where a > 0 and a ≠ 1), the expression a0 = 1. Therefore, loga(1) = 0.
- Example: log10(1) = 0
- Example: log2(1) = 0
- Example: ln(1) = 0
2. Why is log 1 equal to 0?
The value of log 1 is 0 because any non-zero number raised to the power 0 equals 1. In logarithmic form:
- If ax = 1, then x = 0
- So, loga(1) = 0
3. What is the value of log 1 to base 10?
The value of log10(1) is 0. Since 100 = 1, the logarithm of 1 with base 10 must be 0. This is also called the common logarithm of 1.
4. What is the value of ln 1?
The value of ln(1) is 0. The natural logarithm ln(x) has base e, and since e0 = 1, we get ln(1) = 0. This is an important result in calculus and exponential functions.
5. Is log 1 always 0 for any base?
Yes, log 1 is always 0 for any base greater than 0 and not equal to 1. For any valid base a:
- a > 0
- a ≠ 1
- a0 = 1
6. How do you prove that log 1 equals 0?
You prove that log 1 = 0 by converting logarithmic form to exponential form.
- Let loga(1) = x
- This means ax = 1
- Since a0 = 1, we get x = 0
7. What is log 1 in exponential form?
In exponential form, loga(1) = 0 means a0 = 1. Logarithmic and exponential forms are equivalent representations:
- Logarithmic form: loga(1) = 0
- Exponential form: a0 = 1
8. What is the difference between log 1 and log 0?
The key difference is that log 1 = 0, but log 0 is undefined. This is because:
- a0 = 1, so loga(1) = 0
- There is no real number x such that ax = 0 (for a > 0)
9. Can you give an example to calculate log 1?
Yes, for example, log5(1) = 0 because 50 = 1. Step-by-step:
- Let log5(1) = x
- Convert to exponential form: 5x = 1
- Since 50 = 1, x = 0
10. Is log 1 positive, negative, or zero?
The value of log 1 is zero, meaning it is neither positive nor negative. Since loga(1) = 0 for any valid base a, it lies exactly at zero on the number line. This is a standard property of logarithmic functions.
