What Is the Value of Log 10 with Formula Proof and Solved Examples
The concept of value of log 10 plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Whether you are preparing for school exams, JEE, NEET, or any competitive test, knowing the value of log 10 and its properties can simplify your calculations and improve speed and accuracy.
What Is Value of Log 10?
A logarithm tells us the exponent to which a base must be raised to get a certain number. For example, log 10 (with base 10) means “to what power should we raise 10 to get 10?” You’ll find this concept applied in areas such as logarithmic functions basics, exponential and logarithmic conversions, and solving equations in algebra, physics, and chemistry.
Key Formula for Value of Log 10
Here’s the standard formula: \( \log_{10} 10 = 1 \)
It means that when 10 is raised to the power of 1, the result is 10.
Similarly, loge 10 (also written as ln 10) equals approximately 2.302585.
Why Is Value of Log 10 Equal to 1?
The value of log 10 in base 10 is 1 because:
- By definition, \( \log_{a} a = 1 \) for any positive a not equal to 1.
- For \( \log_{10} 10 = x \), we convert to the exponential form:
\( 10^{x} = 10 \) - Clearly, x = 1.
Step-by-Step Illustration
- Given: \( \log_{10} 10 = x \)
- Rewrite in exponential form: \( 10^{x} = 10 \)
- Since \( 10^{1} = 10 \), it follows that \( x = 1 \)
- So, \( \log_{10} 10 = 1 \).
Log 10 Values Table (From 1 to 10)
| Number (N) | log10(N) |
|---|---|
| 1 | 0 |
| 2 | 0.3010 |
| 3 | 0.4771 |
| 4 | 0.6020 |
| 5 | 0.6989 |
| 6 | 0.7781 |
| 7 | 0.8451 |
| 8 | 0.9031 |
| 9 | 0.9542 |
| 10 | 1 |
How to Calculate Other Log Values Using Value of Log 10
If you know the value of log 10, you can easily calculate other log values using logarithm properties:
- For log 100:
log10(100) = log10(102) = 2 × log1010 = 2×1 = 2 - For log 1000:
log10(1000) = log10(103) = 3 × log1010 = 3×1 = 3 - For log210:
Use the change of base formula:
log210 = log1010 / log102 = 1 / 0.3010 ≈ 3.3219
Speed Trick or Vedic Shortcut
Whenever you see log10(10n), just remember – the answer is n, because log1010 = 1. This saves time in MCQs and quick calculations. Practice this and related tricks with Vedantu’s Maths Tricks page.
Try These Yourself
- Find the value of log101000.
- Calculate log210 using log102.
- Express log101 in exponential form and solve.
- What is log10 (0.1)?
Frequent Errors and Misunderstandings
- Confusing base e (ln) and base 10 (common log). ln(10) ≈ 2.3026, not 1!
- Forgetting that logaa = 1 only when a > 0, a ≠ 1.
- Calculating log values without checking the base.
- Making calculation mistakes when converting between exponential and logarithmic forms.
Relation to Other Concepts
The idea of value of log 10 connects closely with topics such as exponents and powers and logarithmic functions. Mastering this helps with solving indices, understanding scientific notation, and simplifying equations in advanced mathematics and science.
Classroom Tip
A quick way to remember value of log 10 is the rule: “log base 10 of 10 is always 1.” Visual log tables, color codes, and practice with log identities during revision classes make it easier. Vedantu’s teachers often use such visual aids and example-based learning to simplify this topic during live classes.
We explored value of log 10—from its clear definition, standard formula, stepwise proofs, example problems, and common mistakes to connections with other log concepts. Continue practicing with log values table, log tables, and logarithm questions on Vedantu to become confident in solving log problems quickly and accurately.
FAQs on Value of Log 10 in Base 10 Explained Clearly
1. What is the value of log 10?
The value of log 10 (base 10) is 1 because 10 raised to the power 1 equals 10. In logarithmic form:
log₁₀(10) = 1
This follows from the definition of logarithms:
- If ax = b, then logₐ(b) = x.
- Since 10¹ = 10, the logarithm equals 1.
2. Why is log 10 equal to 1?
The value of log 10 is 1 because 10 must be raised to the power 1 to get 10. By definition:
- log₁₀(10) asks: “To what power must 10 be raised to obtain 10?”
- Since 10¹ = 10, the answer is 1.
3. What is the value of log 10 in base e (natural log)?
The value of ln(10) (log 10 in base e) is approximately 2.3026. This is because:
- ln(10) = logₑ(10)
- Using a calculator, ln(10) ≈ 2.3026
4. What is the difference between log 10 and ln 10?
The difference is that log 10 usually means base 10, while ln 10 means base e. Specifically:
- log₁₀(10) = 1
- ln(10) ≈ 2.3026
5. How do you evaluate log 10 on a calculator?
To evaluate log 10, press the log button and enter 10, which gives 1. Steps:
- Press the log key (base 10).
- Enter 10.
- The display shows 1.
6. What is the general rule for log of a number with the same base?
The general rule is logₐ(a) = 1 for any positive base a ≠ 1. This is because:
- a¹ = a
- Therefore, the logarithm equals 1.
7. What is the value of log 1 and how is it related to log 10?
The value of log₁₀(1) is 0 because 10 raised to the power 0 equals 1. In comparison:
- log₁₀(1) = 0
- log₁₀(10) = 1
8. Can you give an example using log 10 in an equation?
An example equation is log₁₀(x) = 1, which gives the solution x = 10. Steps:
- Convert to exponential form: 10¹ = x
- Simplify: x = 10
9. What are common logarithms and how does log 10 fit in?
A common logarithm is a logarithm with base 10, and log 10 is a direct example of it. In standard notation:
- log(x) usually means log₁₀(x)
- So log(10) = 1
10. What is the change of base formula for log 10?
The change of base formula is logₐ(b) = log(b) / log(a) or ln(b) / ln(a). For log 10:
- log₁₀(10) = ln(10) / ln(10)
- This simplifies to 1.
