Value of log 1 can be evaluated using the logarithm function, which is one of the important mathematical functions. It is commonly used to solve many lengthy problems and reduce the complexity of the problems by reducing the operations from multiplication to addition and division to subtraction. We are providing below here the logarithm values of 1 to 10, which are mostly used in calculations;

log 1 = 0

log 2 = 0.3010

log 3 = 0.4771

log 4 = 0.6020

log 5 = 0.6989

log 6 = 0.7781

log 7 = 0.8450

log 8 = 0.9030

log 9 = 0.9542

log 10 = 1 Â Â

To find the value of logarithm 1, we are going to use one of the properties, defined for the log. Letâ€™s discuss first the properties of the logarithm.

- log(a Ã— b) = log (a) + log (b)
- log(a Ã· b) = log (a) – log (b)
- log(a
^{b}) = b Ã— log (a)

Generally, the logarithm is classified into two types. They are

- Common Logarithmic Function (represented as log)
- Natural Logarithmic Function (represented as Ln)

The log function with base 10 is called the common logarithmic functions and the log with base e is called the natural logarithmic function.

The logarithmic function is also defined by,

if log_{a}b = x, then a^{x} = b. |

Where x is defined as the logarithm of a number â€˜bâ€™ and â€˜aâ€™ is the base of the log function that could have any base value, but usually, we consider it as â€˜eâ€™ or â€˜10â€™ in terms of the logarithm. The value of the variable â€˜aâ€™ can be any positive number but not equal to 1 or negative number.

Now let us find out the logarithm value of 1 with the help of logarithm definition.

## What is the Value of Log 1 or Ln 1?

From the definition of logarithm function, we know,

If log_{a}b = x

Then, a^{x} = b

To find the value if log 1, since the base is not defined here, let us consider the base as 10. Hence, we can write log 1 as log_{10} 1.

Now, from the logarithm definition, we have the value of a = 10 and b = 1. Such that,

log_{10} x = 1

By the logarithm rule, we can rewrite the above expression as;

10^{x }= 1

As we know, any number raised to the power 0 is equal to 1. Thus, 10 raised to the power 0 makes the above expression true.

so, 10^{0} = 1

This will be a condition for all the base value of log, where the base raised to the power 0 will give the answer as 1.

Therefore, the value of log 1 for any base value is zero.

Or

log_{a}1 = 0 |

where a could be any positive value apart from 1.

Similarly, we can represent the natural logarithm value of 1,i.e. Ln 1.

Ln (b) = log_{e }(b)

âˆ´ Ln(1) = log_{e}(1)

Or e^{x} = 1

âˆ´ e^{0} = 1

Hence, **Ln(1) = log _{e}(1) = 0**

In the same way, if we have to find out the value of log 0, then following the same procedure, we can mention the logarithm definition as;

Log_{10}0 = x or 10^{x} = 0

Since 10^{0 } gives the value of 1. That means to get the value of 10^{x} as 0, we have to put x as the least number less than 0, which we can consider to reach till infinity. Such that,

x = -1, -2, -3, -4, -5,……………., -âˆž

If we put, x as -1, we get, 10^{-1}= 1/10 = 0.1. In the same way, if we put the other values like -2,-3,-4,-5,…., for x, we get the output as,0.01, 0.001, 0.0001 and so on.

So, we can consider, 10 raised to the power -âˆž will give the output as 0.

Therefore, 10^{-âˆž }= 0

Or log_{10}0 = -âˆž

Or log_{a}0 = -âˆž

**Example: **Solve log 1 – log 0

**Solution:** Given, log 1 + log 0

log 1 = 0 and log 0 = -âˆž

Therefore, log 1 + log 0 = 0 -(-âˆž) = âˆž

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