Log e Explained: Formula, Laws, and Exam Questions
The concept of value of log e plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios, especially in algebra, calculus, and competitive tests.
What Is Value of log e?
The value of log e usually means the logarithm of Euler’s number “e” (where e ≈ 2.71828). If no base is mentioned, log often refers to base 10 (common logarithm). So, log10(e) gives the value of log e in base 10, and loge(e) (called natural log or ln e) is in base e. You’ll find this concept applied in algebraic simplifications, physics equations, and computer algorithms.
Key Formula for Value of log e
Here are the standard formulas used most often:
| Expression | Base | Value | Rounded Value |
|---|---|---|---|
| loge(e) | e | 1 | 1 |
| log10(e) | 10 | 0.4342944819 | 0.434 |
| ln(e) | e | 1 | 1 |
Cross-Disciplinary Usage
The value of log e is not only useful in maths but also plays an important role in physics (such as exponential decay and growth), chemistry (reaction rates), computer science (algorithms and complexity), and statistics (probability distributions). Students preparing for JEE, NEET, or board exams will find it commonly used in various problems.
Step-by-Step Illustration
Let’s solve for log10(e) step by step:
1. Use the change of base formula:2. log10(e) = ln(e) / ln(10)
3. Since ln(e) = 1 and ln(10) = 2.302585...,
4. log10(e) = 1 / 2.302585 = 0.4342944819
5. **Final Answer:** log10(e) ≈ 0.434
Now, let’s check loge(e):
1. By logarithm properties, loga(a) = 1 for any valid base a.2. Here, a = e, so loge(e) = 1.
3. **Final Answer:** loge(e) = 1
Speed Trick or Vedic Shortcut
Here’s a quick shortcut to remember when dealing with the value of log e.
- If you see loge(e) or ln(e), immediately write the answer as 1.
- If you see log10(e), just use 0.434 for fast calculations in MCQs or mental maths.
In competitive exams like NTSE, JEE, and Olympiads, remembering these standard values saves time on log-based questions. Vedantu’s expert tips often include such rapid recall points as part of live classes.
Try These Yourself
- Calculate loge(e2).
- What is ln(√e)?
- Convert ln(10) to log10(10).
- If log10(e) = 0.434, what is log10(e5)?
Frequent Errors and Misunderstandings
- Confusing log10 and loge (ln).
- Thinking loge(e) = 0 (It is always 1).
- Putting log10(e) = 1 (It should be 0.434).
- Using wrong conversion: ln(x) ≠ log10(x).
Relation to Other Concepts
The idea of the value of log e connects closely with exponents and powers and logarithms in general. Mastering it will help you solve questions in calculus (derivatives involving log or exponential functions) and in topics like log tables used for quick calculations.
Classroom Tip
A quick way to remember: “log base number of itself is always 1.” So, loge(e) = 1 and log10(10) = 1. For base 10 log of e, ‘434’ is your instant recall key for fast answers. Vedantu’s teachers repeat this during concept drills for strong retention.
We explored the value of log e — its definitions, formulas, quick facts, mistakes to avoid, and its deep connection with other maths concepts. Practice these steps regularly and refer to Vedantu’s resources to master logs for all exams!
FAQs on What is the Value of Log e?
1. What is the value of log e?
The value of log e depends on the base of the logarithm. If the base is 10 (common logarithm), log₁₀(e) ≈ 0.434. If the base is e (natural logarithm), then ln(e) = 1. This is because, by definition, logₐ(a) = 1 for any base a.
2. What is log e base 10?
The common logarithm of e, denoted as log₁₀(e), is approximately 0.4342944819. This value is obtained by using a calculator or by applying the change of base formula: log₁₀(e) = ln(e) / ln(10) ≈ 1 / 2.302585 ≈ 0.434.
3. Why is logₑ(e) = 1?
This is a fundamental property of logarithms. The logarithm of a number to the same base is always 1. In other words, logₐ(a) = 1 for any positive base a (excluding 1). This is because the logarithm asks, “To what power must we raise the base (a) to obtain the number (a)?”. The answer is always 1 (a¹ = a).
4. What is the natural logarithm of e?
The natural logarithm of e, denoted as ln(e) or logₑ(e), is equal to 1. The natural logarithm uses e (approximately 2.71828) as its base.
5. What is the value of e in logarithms?
In logarithms, e represents Euler's number, an irrational mathematical constant approximately equal to 2.71828. It's the base of the natural logarithm (ln).
6. How do you convert between log base e and log base 10?
Use the change of base formula: log₁₀(x) = ln(x) / ln(10) or ln(x) = log₁₀(x) * ln(10). Since ln(10) ≈ 2.303, you can approximate: log₁₀(x) ≈ ln(x) / 2.303 and ln(x) ≈ 2.303 * log₁₀(x).
7. What is log ex?
logₑ(ex) = x. This is because the logarithm and exponential functions are inverse operations with the same base.
8. What are some applications of log e in mathematics?
The natural logarithm (log e or ln) has numerous applications in calculus, including finding derivatives and integrals of exponential functions, solving differential equations, and analyzing exponential growth or decay. It’s also crucial in various mathematical models.
9. What are some applications of log e in physics?
Log e (ln) is frequently used in physics to model exponential processes like radioactive decay, capacitor discharge, and the behavior of certain damped oscillations. It appears in formulas related to thermodynamics, such as entropy calculations.
10. What happens if the base of the log is 1 or less?
Logarithms are undefined for bases less than or equal to 0, and for a base of 1. The base must be a positive number other than 1.
11. Is log e used in competitive exams?
Yes, understanding the properties and applications of log e (particularly natural logarithms) is essential for various competitive exams in mathematics, physics, and engineering. Expect questions involving its applications in calculus, exponential functions, and problem-solving.
12. What are common mistakes students make with log e?
Common errors include: confusing ln(x) with log₁₀(x); misapplying logarithm rules; incorrectly evaluating log e to a base other than e or 10; and forgetting that logₐ(a) = 1.
