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Free online tool that helps you calculate the logarithm of a given base and number.

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In mathematics, a logarithm is an exponent or power to which a given base must be raised in order to obtain a specific number. More formally, if a is a positive real number and b is a positive real number not equal to 1, then the logarithm of b to base a, denoted as log_a(b), is the power to which a must be raised to obtain b.

For example, if we have a base of 2 and a number of 8, then log_2(8) = 3, because 2 to the power of 3 equals 8. Similarly, if we have a base of 10 and a number of 100, then log_10(100) = 2, because 10 to the power of 2 equals 100.

Logarithms are used in various fields of mathematics, science, engineering, and finance. They can help simplify calculations, particularly when dealing with very large or very small numbers. They can also be used to solve equations, analyze data, and model complex systems. The properties of logarithms make them particularly useful for manipulating exponential functions and for studying exponential growth and decay.

Common logarithm and natural logarithm are two different types of logarithms used in mathematics.

The common logarithm, denoted as log, is a logarithm with a base of 10. The common logarithm of a number is the power to which 10 must be raised to obtain that number. The common logarithm is commonly used in everyday calculations, such as in measuring pH and sound levels, and in finance and accounting.

For example, if we want to calculate the common logarithm of 1000, we write log(1000). The value of log(1000) is equal to 3, which means that 10 raised to the power of 3 is equal to 1000 (i.e., 10^3 = 1000).

The natural logarithm, denoted as ln, is a logarithm with a base of e, where e is the mathematical constant approximately equal to 2.71828. The natural logarithm of a number is the power to which e must be raised to obtain that number. The natural logarithm is commonly used in calculus and advanced mathematics, especially in the study of exponential functions and their derivatives.

For example, if we want to calculate the natural logarithm of 10, we write ln(10). The value of ln(10) is approximately 2.30259, which means that e raised to the power of 2.30259 is equal to 10 (i.e., e^2.30259 ≈ 10).

In summary, the main difference between natural logarithm and common logarithm is the base used in the logarithmic expression. The natural logarithm uses the base e, while the common logarithm uses the base 10.