As a mathematician, one of the most important subjects we deal with is the logarithm. The logarithm is a mathematical function that we often use to solve problems in various fields. In this article, we will be examining the process of rewriting the logarithm as a ratio of common logarithms and natural logarithms.

Before we go into the details of how to express the logarithm as a ratio of common logarithms and natural logarithms, we need to understand what a logarithm is. A logarithm is the inverse function of an exponential function. For example, if we have the exponential function y = bx, then the logarithm of y with base b can be expressed as logby = x. In other words, logarithm is the power to which a number must be raised to get another number.

There are two types of logarithms commonly used in mathematics: common logarithms and natural logarithms. A common logarithm is a logarithm with base 10, while a natural logarithm is a logarithm with base e, where e is a mathematical constant approximately equal to 2.71828.

When rewriting the logarithm as a ratio of common logarithms and natural logarithms, we need to use the change of base formula. The change of base formula states that any logarithm with base b can be rewritten as a ratio of logarithms with base a and b, where a can be any positive number except 1. The formula is as follows:

logab = (logcb)/(logca)

where a, b, and c are positive numbers and a and c cannot equal 1.

Now, let us look at an example of how to rewrite the logarithm log52 as a ratio of common logarithms and natural logarithms. First, we will use the change of base formula to rewrite log52 as follows:

log52 = (log102)/(log105)

Next, we will simplify the equation by evaluating the logarithms on the right-hand side. The common logarithm of 10 is 1 and the common logarithm of 5 is 0.69897 (rounded to five decimal places). The natural logarithm of 10 is 2.30259 (rounded to five decimal places) and the natural logarithm of 5 is 1.60944 (rounded to five decimal places). Therefore, we can rewrite log52 as follows:

log52 = (1)/(0.69897) / (1.60944)

Simplifying further, we get:

log52 = 1.51143

Hence, we have successfully rewritten log52 as a ratio of common logarithms and natural logarithms.

In general, when rewriting a logarithm as a ratio of common logarithms and natural logarithms, we should simplify the ratio as much as possible. This means evaluating each of the logarithms and dividing the values to obtain a rational number. If the resulting number cannot be simplified any further, then we have successfully expressed the logarithm as a ratio of common logarithms and natural logarithms.

It is important to note that the change of base formula can be used to rewrite any logarithm as a ratio of logarithms with different bases. This means that we can use the formula to rewrite a logarithm with any base as a ratio of logarithms with any other two bases.

To conclude, rewriting a logarithm as a ratio of common logarithms and natural logarithms is a simple process that involves using the change of base formula. This formula allows us to express any logarithm with any base as a ratio of logarithms with any other two bases. It is important to simplify the ratio as much as possible to obtain a rational number. The ability to rewrite logarithms in this way is an important tool for solving problems in various fields of mathematics.