Why is log 10 equal to 1?
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 Maths – Definition, Formula & Table
1. What is the value of log 10 (base 10)?
The value of log₁₀ 10 is 1. This is because 10 raised to the power of 1 equals 10 (10¹ = 10). This is a fundamental concept in logarithms.
2. Why is log₁₀ 10 equal to 1?
By definition, the logarithm base b of a number x (logbx) is the exponent to which b must be raised to produce x. In this case, 10 raised to the power of 1 equals 10, hence log₁₀ 10 = 1.
3. What is the difference between log 10, log base 10, and ln 10?
• log 10 typically refers to the common logarithm, meaning the base is 10 (log₁₀ 10 = 1).
• log base 10 explicitly states the base is 10, identical to the common logarithm.
• ln 10 represents the natural logarithm of 10, meaning the base is the mathematical constant e (approximately 2.718). ln 10 ≈ 2.303.
4. How do I calculate the value of log 100 (base 10)?
Using the logarithm power rule, log₁₀ 100 = log₁₀ (10²) = 2 log₁₀ 10 = 2 * 1 = 2.
5. How do I calculate the value of log 1000 (base 10)?
Similarly, log₁₀ 1000 = log₁₀ (10³) = 3 log₁₀ 10 = 3 * 1 = 3.
6. What is the value of log 1 (base 10)?
Any logarithm of 1, regardless of the base, is always 0. This is because 10⁰ = 1.
7. What is the relationship between logarithms and exponents?
Logarithms and exponents are inverse operations. If bx = y, then logb y = x. They essentially undo each other.
8. How is log 10 used in solving equations?
Logarithms are frequently used to solve equations involving exponents. They allow us to bring down exponents, making the equations easier to manipulate and solve.
9. What are some common applications of logarithms?
Logarithms have numerous applications, including:
• Chemistry: Calculating pH values.
• Physics: Measuring sound intensity (decibels).
• Finance: Calculating compound interest.
10. How can I use a log table to find log values?
Log tables provide pre-calculated logarithm values for a range of numbers. To find a value, locate the number in the table and read its corresponding logarithm.
11. What is the approximate value of ln 10?
The natural logarithm of 10 (ln 10) is approximately 2.303.
12. How do I convert between exponential and logarithmic forms?
If bx = y, then the equivalent logarithmic form is logb y = x. Remember to correctly identify the base (b), exponent (x), and result (y).
