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Log e

Log e is a constant term that gives the value of the logarithmic function log x, when the value of x is equal to e. The value of log e is approximately equal to 0.4342944819 where the base of the logarithmic function is equal to 10. So, we have log₁₀e = 0.4342944819. As we know, 'e' is an exponential constant, also known as the Euler's number, which is an irrational constant whose value is approximately equal to 2.71828.

In this article, we will determine the value of log e (log function with base 10) and the value of ln e, that is, log e base e. We will also discuss the derivative of log e and solve a few examples using the value of log e for a better understanding of the concept.

1.	What is Log e?
2.	Log e Value
3.	Finding Value of Log e Base 10
4.	What is Ln e (Log e Base e)?
5.	Differentiation of Log e
6.	FAQs on Log e

What is Log e?

Log e gives the value of the logarithmic function log x when x = e. 'e' is an Euler's number whose value is irrational and is approximately equal to 2.71828. We generally use a log function with two bases, one with the base 10 which is commonly called the common logarithmic function, and written as log₁₀x or simply log x, and the second one with the base e, which is called the natural logarithmic function, denoted by ln x or log_ex. The value of log e is approximately equal to 0.43429 (rounded up to 5 digits).

Log e Value

The value of log e is approximately equal to 0.4342944819 where log e has a base equal to 10. We can write this value mathematically as log e OR log₁₀e = 0.4342944819. This value can be determined using the properties of exponential function and logarithmic function. Although log e appears to be a variable value, it is a constant as e is an irrational constant (also called the exponential constant). The image given below shows the approximate value of log e rounded up to 10 digits:

log e value

Finding Value of Log e Base 10

Now that we know the value of log e to be equal to 0.4342944819, we will try finding this value using different formulas and properties of the exponential and logarithmic functions. Using the change of base in log function formula, we can write log e OR log₁₀e as log_ee/log_e10. So, we have

log e = log_ee/log_e10

= (ln e) / (ln 10) ----[ln denotes the natural logarithm]

= 1 / ln 10 --- [Because ln e = 1]

= 1 / 2.302585 --- [Because ln 10 is approximately equal to 2.302585]

= 0.43429448 (approx.)

Hence, we have determined the value of log e using the change of base formula. Another way to find the value of log e is using the value of e. We know that the value of e is approximately 2.71828. So, on substituting this value into the expression log x, we have

log e = log₁₀e

= log₁₀(2.71828)

= 0.434294 (approx.) --- [This value is obtained using calculator]

Therefore, the value of log e with base 10 is approximately equal to 0.43429448.

What is Ln e (Log e Base e)?

As we know, ln x is a natural logarithmic function, that is, it has a base equal to e. It can be written as ln x = log_ex. Now, to find the value of ln e, we can write it as ln e = log_ee which is equal to 1 using the property of logarithmic function which states that if the base and the index are equal, then the value of the log function is equal to 1, that is, log_a(a) = 1. Hence, we can say that the value of ln e is equal to 1.

Differentiation of Log e

The derivative of a function gives the rate of change in the function with respect to the variable. As observed in the previous sections, log e is a constant value and the derivative of a constant term is always equal to zero. Therefore, the derivative of log e is equal to 0 and we can write it as d(log e)/dx = 0.

Important Notes on Log e

The value of log e base 10 is approximately equal to 0.4342944819.
The value of log e base e, that is, ln e value is equal to 1.
The derivative of log e is equal to zero as it is a constant.

☛ Related Topics:

Examples Using Log e

Example 1: Calculate the value of 2 log e.

Solution: We know that the value of log e is approximately equal to 0.4342944819. Therefore, to find the value of 2 log e, we will multiply 2 by log e and hence it is given by,

2 log e = 2 × 0.4342944819

= 0.8685889638

Similarly, we can find the value of any multiple of log e by simply multiplying the scalar with the log e value.

Answer: 2 log e = 0.8685889638
Example 2: Evaluate the derivative of log e^x.

Solution: To differentiate log e^x, we will use the chain rule method of differentiation. Therefore, the derivative is given by:

d(log e^x)/dx = d(log e^x)/d(e^x) × d(e^x)/dx

= (1/ln 10 e^x) × e^x --- [The derivative of log x is equal to 1/(x ln 10) as the base of log x is 10 and the derivative of e^x is equal to e^x]

= e^x / e^x (ln 10)

= 1/ ln 10

Answer: d(log e^x)/dx = 1
Example 3: Find the value of x for the equation e^x = 1.3 using the value of ln e.

Solution: To solve e^x = 1.3, we will apply ln both sides of the equation. So, we have

ln e^x = ln (1.3)

⇒ x ln e = ln (1.3) --- [Because ln a^b = b ln a]

⇒ x × 1 = ln (1.3) --- [Because the value of ln e is equal to 1.]

⇒ x = ln(1.3)

= 0.262364 (rounded till 6 digits)

Answer: The required value of x is 0.262364.

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Log e Practice Questions

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FAQs on Log e

What is Log e in Math?

Log e in math is a constant value which is given by the logarithmic function with base 10, written as log x, when x = e. Its value is equal to 0.4342944819 (approx.).

What is the Value of Log e?

The value of log e is approximately equal to 0.4342944819 (rounded till 10 digits).

How to Do You Find the Value of Log e?

We can find the value of log e using the change of base properties of the exponential function and logarithmic function. We also use the value of the Euler's constant e which is given by, 2.71828.

Why Ln e is Equal to 1?

The value of ln e is equal to 1 because the logarithmic property states that log_a(a) = 1, that is, whenever the base and index are equal in a logarithmic function, its value is equal to 1.

What is the Differentiation of Log e?

The derivative of log e is equal to zero as log e is a constant term and we know that the derivative of a constant function is always equal to 1.

What is Natural Log e?

Natural log e is nothing but the log function with base e. It is written as ln e or log_ee and its value is equal to 1.

Is Log e a Constant?

Yes, log e is a constant. Though it seems to be variable because of 'e' but it is a constant term as 'e' is an exponential constant.

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