1.4.2 Common Mathematical Functions

Question 1

Common Mathematical Functions

Answer 1

• The equation for a line and the quadratic equation are useful in chemistry.
• Logarithms are useful for collapsing measurements that are on an extremely large or small scale into a more manageable scale.
- The equation for a line and the quadratic equation are useful in chemistry.
- The equation for a line is y = mx + b. In this equation, m is the slope of the line and b is the y-intercept. The slope is rise over run. In the example to the left, rise (the change in y) is 0.8 over a run (change in x) of 2.0, so the slope is 0.8/2.0 or 0.4. The y-intercept is the value of y when x equals zero. In the example to the left, the y-intercept is 1.2.
- The quadratic equation is y = ax^2 + bx + c. The graph shown is for the quadratic equation y = x^2 + 2x – 3. The points where the parabola described by a quadratic equation crosses the x-axis can be found using the quadratic formula. Since the quadratic formula includes both a positive and negative root, it will always have two solutions. However, in general only one of the solutions will have a meaning in chemistry.
- Logarithms are useful for collapsing measurements that are on an extremely large or small scale into a more manageable scale.
- A logarithm is the power to which a base must be raised to obtain a given result. For example, if 10^x
= a, x is the base-10 logarithm of a. The base-10
logarithm is written log10.
- Another type of logarithm is the natural logarithm,
abbreviated ln. The natural logarithm is the base-e logarithm, where e = 2.718281829… The natural logarithm appears in many natural relationships.

Question 2

Find the ln (1.3255 × 10^4 ). Then, if 0.28 is the ln of a, what is a ?

Answer 2

• 9.49213; 1.3
• To solve, the ln (13,255) = 9.49213 which correctly has five digits after the decimal (because the data value 13,255 has five significant digits). For the second part, e 0.28 = 1.3231 = 1.3 when rounded to two significant figures (because the 0.28 value has two significant digits).

Question 3

A line can be described by the equation y = mx + b. Which statement best describes the role of each item in this equation?

Answer 3

• x and y are variables, m is the rise / run, b is the y-intercept

Question 4

If 0.45 is the log 10 of a, what is the value of a ?

Answer 4

• 2.8
• If you input 0.45 into your calculator and use the 10x function to find your answer, you will get an answer of 2.818. You then have to account for the number of digits after the decimal (two) in 0.45, and change your answer to 2.8, with two significant digits.

Question 5

What is the base-10 logarithm of 10,000?

Answer 5

• 4

Question 6

The equation for a parabola is y = ax 2 + bx + c. When y = 0, there are two values for x. If a = 1, b = −6, and c = 8, then what are the two values for x ?

Answer 6

• 4, 2

Question 7

Find the log (15,000). Then, if 0.59 is the log 10 of a, what is the value of a ?

Answer 7

• 4.2; 3.9
• To solve, the log 10 (15,000) = 4.176 which, when rounded to two significant figures (because of 15,000) = 4.2. For the second part, 100.59 = 3.89 = 3.9 when rounded to two significant figures (because of the 0.59 value).

Question 8

Suppose that you have the equation y = 3x + b. If one point on this line has the x and y coordinates (2, 7), what is the value of b in this formula?

Answer 8

• b = 1

Question 9

The general equation for a line is y = mx + b. A particular line has the equation y = (1/2) x + b. One point on the line has the coordinates (2, 4). What is the value for b in the equation of the line? Using your value for b, also find the value of y when x = 4.

Answer 9

• 3; 5
• You solve for b by using the (2, 4) data:
  y = (1/2) x + b, so 4 = (1/2) (2) + b, or 4 = 1 + b.
  b = 3
  For the second part, substitute the values for x (4) and for b (3) into the equation and solve for y:
  y = (1/2) (4) + 3 = 2 + 3 = 5.

Question 10

Using the equation y = x^ 2 − 3x − 7, what is the value for the expression b^ 2 − 4ac ?

Answer 10

• 37

Question 11

The term rise/run can also be described as the (change in y) / (change in x). In addition, it is also referred to as the slope of the line. In order to calculate it using two points on a line, you calculate ( y2 − y1 ) / (x2 − x1 ). Suppose you had the following two ordered pairs from a line graph: (1, 1) and (3, 5). What would the rise / run (or slope) of the line be?

Answer 11

• 2