**Solution:**

To find the base of the isosceles triangle, we can use the formula for the area of a triangle, which is given by:

Area = (1/2) * base * height

In this case, we are given the area of the triangle as 240 cm^2. Let's denote the base of the triangle as 'b' and the height as 'h'. Since the triangle is isosceles, we know that the lengths of the two equal sides are 26 m each.

Let's proceed with the solution step by step:

**Step 1: Finding the height of the triangle**

To find the height of the triangle, we can rearrange the area formula:

240 cm^2 = (1/2) * b * h

Multiply both sides of the equation by 2:

480 cm^2 = b * h

**Step 2: Finding the relationship between the base and height**

Since the triangle is isosceles, the height of the triangle bisects the base, dividing it into two equal parts. Let's denote each part of the base as 'x'. Therefore, the base 'b' can be expressed as:

b = 2x

**Step 3: Substituting values and solving the equation**

Substituting the value of 'b' in terms of 'x' in the area equation:

480 cm^2 = (2x) * h

Dividing both sides of the equation by 2:

240 cm^2 = x * h

**Step 4: Solving for 'x' using Pythagorean theorem**

We can use the Pythagorean theorem to find the value of 'x'. Since the isosceles triangle has two equal sides of length 26 m, and the height 'h' bisects the base, we can form a right-angled triangle. Let's denote the hypotenuse as '26 m', one of the equal sides as 'x', and the height as 'h'. Using the Pythagorean theorem, we have:

x^2 + h^2 = 26^2

Substituting the value of 'h' from the previous equation:

x^2 + (240/x)^2 = 26^2

Simplifying the equation:

x^4 - 26^2 * x^2 + (240)^2 = 0

**Step 5: Solving the quadratic equation**

Solving the quadratic equation for 'x', we can use the quadratic formula:

x = [-b ± √(b^2 - 4ac)] / 2a

For our equation, the coefficients are:

a = 1
b = -26^2
c = (240)^2

Substituting these values into the quadratic formula, we can calculate the two possible values of 'x'.