Why is the Value of Log 1 Always Zero? Explained with Formula and 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 Value of Log 1: Meaning, Formula, and Uses in Maths
1. What is the value of log 1 in mathematics?
The value of log 1 is always 0, no matter which base is used (except base 1 or negative bases). This is because any nonzero number raised to the power 0 equals 1: $a^0=1$ where $a$ is the base.
2. Why does log 1 equal zero?
We know logarithms ask, 'To what power must the base be raised to give 1?' Since any base $a$ (where $a>0$ and $a eq 1$) to the power 0 gives 1 ($a^0=1$), log 1 equals 0.
3. Does the value of log 1 change with different bases?
No, the value of log 1 is always 0, regardless of the base (as long as it’s a valid logarithmic base: $a>0$, $a\neq1$). This is true for log base 10, log base $e$, and log base 2.
4. What is the value of log10 1?
The logarithm base 10 of 1, written as $\log_{10} 1$, is 0. This is because $10^0=1$. So, log base 10 of 1 always equals 0 in mathematics.
5. What is the value of loge 1 or ln 1?
The natural logarithm of 1, or $\ln 1$, is 0. This is because $e^0=1$, so the exponent needed to get 1 from base $e$ is 0. Therefore, the ln 1 value is 0.
6. How do you prove that log 1 is zero?
To prove log 1 is zero, use the basic logarithm definition: if $\log_a 1 = x$, then $a^x=1$. Since any number to the power of 0 equals 1, $x=0$. Thus, $\log_a 1 = 0$.
7. Can log 1 be negative or undefined?
The logarithm of 1 is never negative or undefined for any positive base except 1. For base 1 or negative bases, the logarithm is not defined. For all valid bases, log 1 is always 0.
8. What is the general logarithm formula for log 1?
The general logarithm formula for log 1 is $\log_a 1 = 0$, where $a$ is any valid base ($a>0$, $a\neq1$). This holds true for all standard logarithmic operations in mathematics.
9. How is log 1 used in solving equations?
The value of log 1 helps solve equations by simplifying expressions. For example, in a logarithmic equation, if you have $\log_a x = 0$, then $x = 1$. This property makes it easier to find solutions quickly.
10. What are common mistakes when calculating log 1?
Common mistakes with log 1 include:
- Using an invalid base (like 1 or negative numbers)
- Assuming log 1 is undefined
- Incorrectly claiming log 1 can be negative
11. Is log 1 used in real-life calculations?
Yes, the value of log 1 is often used in real-life calculations, especially in science and engineering. It helps simplify logarithmic equations and signals where variables return to their starting or reference value.
