## Inverse of an exponential function and Graphs of Logarithmic Functions

Any positive real number can be the exponent of a power by drawing the graph of the exponential function for 0 < b < 1. Since is a one-to-one function, its reflection in the line y = x is...

## Solve linear equations with rational coefficients by multiplying by the least common denominator to clear the fractions

Often when solving linear equations we will need to work with an equation with fraction coefficients. We can solve these problems as we have in the past. This is demonstrated in our next example. Example 1: Find the solution...

## Two–Dimensional Geometry Problems

Most geometry problems are composed of rectangles, triangles, and circles. It is therefore useful to review the formulas for the perimeter and area of these standard geometric shapes. A rectangle of length l and width w has a perimeter...

## Solve one step linear equations by balancing using inverse operations

Solving linear equations is an important and fundamental skill in algebra. In algebra, we are often presented with a problem where the answer is known, but part of the problem is missing. The missing part of the problem is...

## The possible partition into some parts

Solution IMO 2015 (combinatoric shortlist ) Problem 7: In a company of people some pairs are enemies. A group of people is called unsociable if the number of members in the group is odd and at least 3, and...

## Proving that there infinitely positive integer that are not clean (unique representation)

Solution IMO 2015 (combinatoric shortlist problem ) Problem 6: Let S be a nonempty set of positive integers. We say that a positive integer n is clean if it has a unique representation as a sum of an odd...