How to Use Log Tables for Fast Calculation in Maths
In Mathematics, the logarithm is the most convenient way to express large numbers. The definition of the logarithm can be stated as the power to which any number must be raised to obtain some values. Logarithms are also said to be the inverse process of exponentiation. In this article; we will study Logarithm functions, properties of logarithmic functions, log value table, the log values from 1 to 10 for log base 10 as well as the log values from 1 to 10 for log base e.
Log values are important in mathematics and other related subjects such as physics. Students need to refer to the log values for finding different sums related to logarithms. The value of log 1 to the base 10 is given zero. The log values can be determined by using the logarithm function. There are different types of logarithmic functions. Log functions are useful for finding lengthy calculations and saving time. Using a logarithm function also makes it easier to solve a complex problem. By using logarithm functions students can reduce the operations from multiplication to addition and division to subtraction. Read here to know more about logarithm functions.
Logarithms Function
The logarithm function is defined as an inverse function of exponentiation.
Logarithms function is given by
F(x) = loga x
Here, the base of the logarithm is a. It can be read as a log base of x. The most commonly used logarithm functions are base 10 and base e.
Rules for Logarithm
There are some rules of logarithm and students must know these rules to solve questions. The rules are given here:
Common Logarithms Function-
The logarithm function with base 10 is known as Common Logarithms Function. It is expressed as log10.
F(x) =log10 x
Natural Logarithms Function -
The logarithm function with base e is known as Natural Logarithms Function. It is expressed as loge.
F(x) =loge x
Product Rule
In the product rule, two numbers will be multiplied with the same base and then the exponents will be added.
Logb MN = Logb M + Logb N
Quotient Rule
In the quotient rule, two numbers will be divided with the same base and then the exponents will be subtracted, Logb M/N = Logb M - Logb N
Power Rule
In the power rule, exponents' expressions are raised to power and then the exponents are multiplied.
Logb Mp = P logb M
Zero Exponent Rules
Loga = 1
Change of Base Rule
Logb (x) = in x/ In b or logb (x) = log10 x / log10 bValue of Log 1 to log 10 for Log Base 10 Table
Log Table 1 to 10 for Log Base 10
Here, we will list the log values from 1 to 10 for loge e in tabular format.
Log Table 1 to 10 for Log Base e
How to find the value of Log 1?
According to the definition of logarithm function, logan=x can be written as an exponential function:
Then ax = b
When the value of log 1 is not given, you can take the base as 10. Thus, you can express it as log 1 as log10 1.
Now, according to the definition of logarithm, we know the value of a =10 and b =1. Thus,
Log 10 x = 1
We can also write this as:
10x= 1
We already know that anything raised to the power 0 is equal to 1. Thus, 10 raised to the power 0 will tell that the above given expression is true.
So, 100= 1
This is the general condition for the base value of log and the base raised to the power zero will give you the value of 1.
This proves that the value of log 1 is 0.
Alternative method to find log 1 or log to the base e?
We can also find the log value of 1
Log (b) = loge (b)
Thus, Ln(1) = loge(1)
Or ex = 1
∴ e0 = 1
Hence, Ln(1) = loge(1) = 0
Important points to remember
Students must remember a few important points related to the logarithms. Some important points to remember are:
India was the first country in the 2nd century BC to use logarithm
Logarithm was first used in contemporary times by a German mathematician named Michael Stifel.
The inverse process of logarithms is also known as exponentiation
If one has to do theoretical work, natural logs are the best. They are easy to figure out quantitatively.
The most important advantage of using base 10 logarithms is that they are easy to calculate mentally for some numbers. For example, the log base 10 of 1,00,000 is 5 and you only have to count the zeroes.
Solved Examples
Solve the Following for the Value of x for log3 x = log34 + log37 by using the Properties of a Logarithm?
Solution: log3x = log34 + log37
= log34 + log37 = log3 (4 x 7) (by using the addition rule)
= log3(28)
Hence, x = 28
Evaluate: log1 – log 0
Solution: log1 – log 0 (Given)
Value of Log 1 = 0 and Value of log 0 = - ∞
Hence, log 1+ log 0 = 0-(-∞) = ∞
Find the value of log2(64)
Solution: x =64 (Given)
By using the base formula,
Log2 x = log10 x/ log10 2
= log2 64 = log10 64/ log10 2
=1.806180/ 0.301030= 6
Quiz Time
1. Logarithm Functions are the Inverse Exponential of
a. Verses
b. Functions
c. Numbers
d. Figures
2. How will you write the Equation 53= 125 in log form
a. Log 3 (125) =5
b. Log 125 (5) = 3
c. Log 5 (125) = 3
d. Log 5 (3 = 124)
3. What will be the value of log 9, if log 27 = 1.431?
a. 0.934
b. 0.945
c. 0.954
d. 0.958
FAQs on Logarithm Values From 1 to 10: Complete Table
1. What are the standard values of log 1 to 10 for base 10?
The common logarithm (base 10) values for the integers from 1 to 10 are fundamental in many calculations. They are:
- log(1) = 0
- log(2) = 0.3010
- log(3) = 0.4771
- log(4) = 0.6021
- log(5) = 0.6990
- log(6) = 0.7782
- log(7) = 0.8451
- log(8) = 0.9031
- log(9) = 0.9542
- log(10) = 1
2. Why is log 1 always equal to 0 for any valid base?
The value of log 1 is always zero because a logarithm is the inverse of an exponent. The expression logb(1) = x asks, "To what power (x) must the base (b) be raised to get 1?" In exponential form, this is written as bx = 1. According to the rules of exponents, any non-zero number raised to the power of 0 is 1. Therefore, x must be 0, making log 1 = 0 for any valid base.
3. What is the main difference between common logarithms (log) and natural logarithms (ln)?
The primary difference lies in their base.
- Common Logarithm (log): Uses base 10. It is widely used in measurement scales that cover a vast range of values, such as the pH scale for acidity and the Richter scale for earthquake intensity.
- Natural Logarithm (ln): Uses base 'e', where 'e' is Euler's number (approximately 2.718). It is essential in calculus, physics, and finance for modelling continuous growth and decay processes.
4. Why are the logarithms of zero and negative numbers undefined?
The logarithm of zero or a negative number is undefined for real numbers. This is because a logarithm, logb(x), answers the question: "what exponent on base 'b' gives 'x'?" Since the base 'b' is always a positive number, raising it to any real power will always result in a positive number. There is no real exponent that can make a positive base equal to zero or a negative number. Thus, the domain of a logarithmic function is restricted to positive numbers only (x > 0).
5. How can you use logarithm properties to find the value of log 50?
You can find the value of log 50 by breaking it down using the fundamental properties of logarithms, specifically the Product Rule. The Product Rule states that log(M × N) = log(M) + log(N).
- First, express 50 as a product of numbers whose logs you know, like 5 and 10: log(50) = log(5 × 10).
- Apply the Product Rule: log(5) + log(10).
- Substitute the known values: 0.6990 + 1.
- Calculate the result: log(50) = 1.6990.
6. In which real-world fields are logarithms most commonly applied?
Logarithms are crucial for simplifying the measurement and calculation of quantities that vary over a very wide range. Key applications include:
- Chemistry: To measure the acidity or alkalinity of a solution using the pH scale, where pH = -log[H+].
- Seismology: To measure earthquake intensity on the Richter scale, where each whole number increase represents a tenfold increase in measured amplitude.
- Acoustics: To measure sound intensity in decibels (dB), which compares sound levels on a logarithmic scale.
- Finance: To calculate compound interest and model economic growth over time.
