**The Third Proportional**

In mathematics, the concept of proportion is used to compare two ratios or fractions. If two ratios are in proportion, it means that they are equivalent to each other. A proportion can be represented as a/b = c/d, where a, b, c, and d are numbers.

One important concept related to proportions is the third proportional. The third proportional is a number that is related to two given numbers in the same way that the second number is related to the first. In other words, if a/b = c/d, then the third proportional to a and b is the number x that satisfies b/x = c/d.

**Example:**

Let's consider the given numbers 12 and 18. We want to find the third proportional to these numbers.

We can set up a proportion: 12/18 = x/18.

To solve for x, we can cross-multiply and get 12 * 18 = 18 * x.

216 = 18x.

Dividing both sides by 18, we find that x = 12.

Therefore, the third proportional to 12 and 18 is 12.

**Relation to Ratio and Proportion:**

The concept of the third proportional is closely related to ratio and proportion. When two ratios are in proportion, we can find the third proportional using the same principles of setting up and solving a proportion.

In the example above, the ratios 12/18 and 18/18 are in proportion, as they are equivalent. By setting up a proportion using these ratios, we can find the third proportional.

**Relation to Indices:**

Indices, also known as exponents, are used to represent repeated multiplication of a number by itself. The concept of the third proportional does not directly relate to indices.

**Relation to Logarithms:**

Logarithms are the inverse operations of exponentiation. They can be used to solve exponential equations and manipulate numbers in a way that simplifies calculations. The concept of the third proportional does not directly relate to logarithms.

In conclusion, the third proportional is a concept related to ratio and proportion. It is used to find a number that is related to two given numbers in the same way that the second number is related to the first. It does not directly relate to indices or logarithms.