Elementary arithmetic, with brief notices of its history... by Robert Potts.

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5. The common logarithm of one number is greater by unity than
the logarithm of another, and the logarithm of their difference is
one-half of the logarithm of their sum. Find the numbers.
6. The area of a right angled triangle whose hypothenuse is
xT3 and sides xT2 and,x, is -2-/2+/ 5.
XII.
1. Define what is meant by the characteristic and mantissa of a
logarithm. Shew that the characteristics of the logarithms of numbers
to base 10 may always be determined by inspection, and that the
mantissa depends on the sequence of figures only, and not on the
position of the decimal point.
2. Explain why the mantissa of a logarithm is always in calculation considered positive, while the characteristic may be positive or
negative.
3. Given log,081-123= 9096256, write the logarithms of 81123,
-81123, '0081123, 8112300, and 8112-3 respectively.
4. Find the characteristics of the logarithms of 300, '003, and 3-2
to base 5; of 81 to base 3; of 81 to base /3; of 25 to base 100; of
~003 to base 8.
5. Having given log102= 30103, log103= 47712, logo07 =84509;
find the characteristics of log121200, logo17-1; log4 /3)0; log,3 49:
lgooo,000 -}4
6. Find the number of digits in 263, 125100, 100010CO, e~00~ respectively,
having given log0,2 == 3010300 and logloe = 4 342956.
7. If 5-lO~~~0 be expressed as a decimal, how many ciphers will
there be before the first significant digit of the decimal?
8. Find the integral values between which x must lie, in order
that the integral part of (1-08)' may contain 4 digits.
9. How many times must the number 2 be multiplied into itself,
so that the product may just exceed 1,000,000,000?
10. Given p the characteristic of logaZV, what is the characteristic
of log_,N, when x = az?
11. If the characteristics of the logarithms of any numbers be
known, find the limits between which the characteristic of the logarithm of their product will lie; (1) if all the numbers be greater than
unity, and (2) if a certain number of them be less than unity.
XIV.
1. What is meant by the nodulus of any two systems of logarithms?
Find a general expression for the modulus which connects any two
systems of logarithms, and apply it to find the moduli which connect
the systems whose bases are 9 and 10; 10 and 20; and 5 and 7
respectively.
2. Prove that the ratio logex is a constant quantity, and incomlogx