Chapter 17 – Logarithms to Base 10 (L Scale)

Recall the Laws of Logarithms:

And the equivalent Logarithmic and Exponential form:

and

17.1 Logarithms and Antilogarithms Using L and D scales.

(i.e. usual L scale on body of the slide rule)

Fig 17-1

Example 1: (Fig. 17-1)

(Or this could be stated )

  1. Set the hair line over 1.82 on the D scale.
  2. Under the hair line read off 0.26 on the L scale as the answer.

Note:

  1. For numbers between 1 and 10 on the D scale, their logarithms are read directly off the L scale as the values between 0 and 1.
  2. For the logarithms of numbers outside the range 1 to 10, we have to decide their characteristic ourselves.

    3. e.g. or

    The Slide Rule gives us only the mantissa, as do logarithm tables.

  3. If we have the logarithm of a number and wish to find the number, we work the opposite way, (i.e. from the L scale to the D scale.)

Example 2: antilogarithm of 3.26 = 1,820

(or this could be stated )

  1. Set the hair line over 0.26 (i.e. mantissa only) on the L scale.
  2. Under the hair line read off ‘182’ on the D scale as the answer.

(as the characteristic is 3)

(See exercise 17(a) at the end of 17.2 for problems)

Note:

As sines, cosines and tangents are found on the D scale, the value of log sin, log cos and log tan can be obtained by reading from the angle on the appropriate trigonometrical scale directly onto the L scale.

 

17.2 Logarithms and Antilogarithms Using L and W (Root) scales.

This is the system applicable to the Faber-Castell Slide Rules 2/83N, 62/83 etc. The L scale is on the slide and it is used in conjunction with the W scales. It is best to use the W’1 and W’2 scales instead of the W1 and W2 scales to avoid any error, should the slide be slightly displaced.

    1. For a number on the W’1 scale, that is a number less than about 3.2 (3.5 for the 2/83 N), we read its logarithm off the L scale according to the numbers to the left of the graduations.

Example 1:

  1. Set the hair line over 1.385 on the W’ scale.
  2. Under the hair line read off 0.143 on the L scale (according to numbers to left of graduations) as the answer.
    1. For a number on the W’2 scale, that is a number greater than about 3 (2.8 for the 2/83N), we read its logarithm off the L scale according to the numbers to the right of the graduations.

Example 2:

  1. Set the hair line over 82.4 the W’2 scale.
  2. Under the hair line read off 0.916 on the L scale (according to numbers to right of graduations) as the mantissa of the answer.

answer = 1.916 (as 82.4 is between 10 and 100).

Exercise 17(a)

  1. =
  2. =
  3. =
  4. =
  5. =
  6. =
  7. =

    9. Find X in the Following:

 

17.3 Raising Numbers to Powers and Solving Exponential Equations:

A. Raising a Number to Power. (A better method using LL scales is given in unit 19.)

Example 1:

Express as (by Law III)

(Using Slide Rule to Find logarithm)

(Multiply using Slide Rule)

(Use the Slide Rule again to obtain the antilog of 0.55 and position the decimal point according to the characteristic, 7.)

answer = 35,450,000

B. Solving an Exponential Equation.

Example 2: solve 3x = 5 for x

If two quantities are equal, then their logarithms will be equal.

i.e.

(evaluate each using Slide Rule)

(divide using Slide Rule)

(evaluate each using Slide Rule)

Exercise 17(b)

  1. =
  2. =
  3. =
  4. =

    5. Find x in the following:

(Hint, write as )