# Log Calculator Perform logarithmic calculations effortlessly with our Log Calculator. Calculate logarithms with different bases, simplify complex expressions, and solve logarithmic equations accurately. the logarithm equation logbx=y. It can accept "e" as a base input.

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### What is Log?

The logarithm, or log, is the inverse of the mathematical operation of exponentiation. This means that the log of a number is the number that a fixed base has to be raised to in order to yield the number. Conventionally, log implies that base 10 is being used, though the base can technically be anything. When the base is e, ln is usually written, rather than loge. log2, the binary logarithm, is another base that is typically used with logarithms. If, for example:

x = by; then y = logbx; where b is the base

Each of the mentioned bases is typically used in different applications. Base 10 is commonly used in science and engineering, base e in math and physics, and base 2 in computer science.

### Basic Log Rules

When the argument of a logarithm is the product of two numerals, the logarithm can be re-written as the addition of the logarithm of each of the numerals.

logb(x × y) = logbx + logby
EX: log(1 × 10) = log(1) + log(10) = 0 + 1 = 1

When the argument of a logarithm is a fraction, the logarithm can be re-written as the subtraction of the logarithm of the numerator minus the logarithm of the denominator.

logb(x / y) = logbx - logby
EX: log(10 / 2) = log(10) - log(2) = 1 - 0.301 = 0.699

If there is an exponent in the argument of a logarithm, the exponent can be pulled out of the logarithm and multiplied.

logbxy = y × logbx
EX: log(26) = 6 × log(2) = 1.806

It is also possible to change the base of the logarithm using the following rule.

logb(x) =
 logk(x) logk(b)
EX: log10(x) =
 log2(x) log2(10)

To switch the base and argument, use the following rule.

logb(c) =
 1 logc(b)
EX:   log5(2) =
 1 log2(5)

Other common logarithms to take note of include:

logb(1) = 0
logb(b) = 1
logb(0) = undefined
limx→0+logb(x) = - ∞
ln(ex) = x

### Log Calculator Example

Logarithmic Calculation Base Operand Result
log 10 10 100 ?
ln e 2.718 ?
log 5 5 25 ?

In this example, we have three different logarithmic calculations that we want to solve using the Log Calculator.

To solve logarithmic calculations:

1. Identify the base and operand for each logarithmic calculation.
2. Enter the base and operand values into the Log Calculator.
3. The calculator will compute the result of the logarithmic calculation.

Let's solve each logarithmic calculation:

1. For the logarithmic calculation log base 10 of 100:
• Base = 10, Operand = 100.
• Entering these values into the Log Calculator gives the result:
```复制代码```log 10 (100) = 2
``````

Therefore, the logarithm base 10 of 100 is equal to 2.

1. For the logarithmic calculation natural logarithm (ln) of 2.718:
• Base = e, Operand = 2.718.
• Entering these values into the Log Calculator gives the result:
```复制代码```ln (2.718) ≈ 1
``````

Thus, the natural logarithm (ln) of 2.718 is approximately 1.

1. For the logarithmic calculation log base 5 of 25:
• Base = 5, Operand = 25.
• Entering these values into the Log Calculator gives the result:
```复制代码```log 5 (25) = 2
``````

Hence, the logarithm base 5 of 25 is equal to 2.

Using the Log Calculator, you can easily perform logarithmic calculations and obtain accurate results. The table will now look as follows:

Logarithmic Calculation Base Operand Result
log 10 10 100 2
ln e 2.718 1
log 5 5 25 2

Therefore, the results of the logarithmic calculations are as given in the table above.