REVIEW OF BASIC MATHEMATICS

 

The following short review of basic mathematics may be useful to you in solving the problems that are assigned from Applied Hydrogeology.

 

LOGARITHMS

 

The logarithm of a number is the exponent of that power to which a base number must be raised to yield the value of the number. There are two bases. Common logarithms use the base 10 and are designated by log. Natural logarithms use the base 2.718… and are designated by ln. The number 2.718… is also designated by the letter e.

 

 

The log of 25.7 is 1.409933 and may be found by entering the number 25.7 on your calculator and then pressing the “log" key. This means that if you take the number 10 and raise it to the 1.409933 power, you will get 25.7. You can do this on your calculator by

entering 10, pressing the “y^x” key, entering 1.409933 and then pressing the “=” key. This process is known as finding the antilog or inverse log of a number. Thus, if 1.409933 is the log of 25.7, then 25.7 is the inverse log of 1.409933.

 

 

The ln of 25.7 is 3.246491 and may be found by entering the number 25.7 on your calculator and then pressing the "ln x" [or “ln"] key. This means that if you start with e and raise it to the 3.24691 power, you will get 25.7. If you have an “ex" key on your calculator, you can enter 3.246491 and the press the “ex" key to yield 25.7. If you don't have an “ex" key, you can still perform the operation. Since we are finding the inverse ln, enter 3.246491 and press the “INV" key followed by the “ln" [or "ln x"] key.

 

 

Logarithms have the following relationships:

 

log ab = log a + log b

 

log a/b = log a -log b

 

log an = n log a

 

Both the natural and common logarithm of 1.0000 is 0. The logarithm of any number greater than 1.0000 is positive. The logarithm of any number less than 1.0000 is negative.

 

          The log of 10 is 1, the log of 100 is 2, the log of 1000 is 3, etc.

 

 

EQUATIONS FOR AREA

 

Area of a circle with radius r

A =  pr2         (pi = 3.1416)

 

          Area of a triangle with a base b and altitude h

A = 1/2 bh

 

Area of a rectangle with sides a and b

A = ab

 

Area of a parallelogram with sides a and b and an included angle z

A = ab sin z.

 

Area of a trapezoid whose parallel sides are a and b and with an altitude h

A = 1/2(a + b)h.

 

EQUATIONS FOR CIRCUMFERENCE

 

Circumference of a circle with a diameter d

C = p d.

 

Circumference of a triangle of sides a, b and c

C = a + b + c.

 

Circumference of a rectangle with sides a and b

          C  = 2 a +  2b.

 

EQUATIONS FOR VOLUME

 

Volume of a regular prism

V = area of base x altitude.

 

Volume of a pyramid

V= 1/3 area of base x altitude.

 

Volume of a cylinder with radius r and height h

V = p r2 h

 

Volume of a cone with radius r and height h

V = 1/3 p r2 h

 

EQUATION OF A STRAIGHT LINE IN RECTILINEAR COORDINATES

 

y = m x + b

where

m is the slope of the line 

b is the intercept of the y axis.

 

TRIGONOMETRIC FUNCTIONS OF A RIGHT TRIANGLE

 


 

 

 


Sine A = sin A = a/h

Cosine A = cos A = b/h

Tangent A = tan A = a/b

Cotangent A = cot A = b/a

Secant A = sec A = h/b

Cosecant A = csc A = h/a

 

DIFFERENTIALS

 

d ax = a dx

 

d (u + v) = du + dv

 

d uv = udv + vdu

 

d (u/v) = (vdu – udv)/v2

 

d xn = n xn-l dx

 

d xy = yxy-1 dx + xy ln x dy

 

d ex = ex dx

 

d eax = a eaxdx

 

d ax = ax ln a dx

 

d ln x = x -1 dx

 

d log x = x -1 log e dx