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In its simplest form, a logarithm is an exponent. Taking the logarithm of a number, one finds the exponent to which a certain value, known as a base, is raised to produce that number once more.
Logarithms have the following structure:
where b is known as the base, c is the exponent to which the base is raised to afford x.
Consider the following logarithm, for example:
The left side of the equation states that the right will be the exponent to which 3 is raised to yield 243 and indeed, 35=243.
The explanation of the previous example reveals the inverse of the logarithmic operation: exponentiation. Starting with 243, if we take its logarithm with base 3, then raise 3 to the logarithm, we will once again arrive at 243.
Logarithms are most commonly used to simplify complex calculations that involve high-level exponents. In science and engineering, the logarithm with base 10 is commonly used.
In pure mathematics, the logarithm with base e (≈ 2.718) is often applicable. In computer science, the binary logarithm, with base 2, is commonly used.
In natural science and engineering, the logarithm with base 10 is frequently used. In chemistry, for example, pH and pKa are used to simplify concentrations and dissociation constants, respectively, of high exponential value.
The purpose is to bring wide-ranging values into a more manageable scope. A dissociation constant may be smaller than 1010, or higher than 10-50. Taking the logarithm of each brings the values into a more comprehensible scope (10 to -50) .
Interactive Graph: Graph of Binary Logarithm
Graph of binary logarithm with the equation $y=2log(x)$. The function's slope decreases with increasing x, thus containing its vertical growth. More importantly, the scale is greatly reduced, allowing for easy visualization of large values of y. In many cases, by taking the logarithms of exponential equations, the equations become linear, and plotting the appropriate logarithms can help solve for roots, slopes, and intercepts.
A logarithm is an exponent., A logarithm with base e is called the natural logarithm., The inverse of the logarithmic operation is exponentiation., and A logarithm with no base written is assumed to have base 2.