391. In the same system the sum of the logarithms of two numbers is the logarithm of their product; and the difference of the logarithms is the logarithm of their quotient.

Let a = loga n, and

a+y = nn', and a-9 =

; or a + y is loga nn',

and

y

;

n

that is, logann' = logan + logan'; and loga

= log1n - log1n'.

n

Ex. 2.

Ex. 3.

Ex. 4.

Log pqr = log pq + log r = log p + log q + log r.

Log = log 5 – log 7.

Log1006 = log 10 ́

6

Ex. 5. Log10 0006 log10 6-logo 1030-77815 - 3.

=

The last two results are usually written 1.77815, 377815.

392. If the logarithm of a number be multiplied by n, the product is the logarithm of that number raised to the nth power.

Let N be the number whose logarithm is x, or a* = N; then a = N"; that is, na is the log. of N", or log, N" = n. log, N. Exs. Log (13) = 5 x log 13. Log b2 = ≈ × log b. COR. Log (amb" c2 ... ) = m log a + n log b + p log c + ...

:

Ex. Log√a2x2 = log (√a + x . √a − x) = 1⁄2 log a + x + 1⁄2 log a − x. 393. If the logarithm of a number be divided by n, the quotient is the logarithm of the nth root of that number.

394. The utility of a Table of logarithms in arithmetical calculations will from hence be manifest; the multiplication and division of numbers being performed by the addition and subtraction

of these artificial representatives; and the involution or evolution of numbers by multiplying or dividing their logarithms by the indices of the powers or roots required.

Also the value of x which satisfies an equation of the form a* = b, may be found since x. log a = log b, and .. x=

log b log a

But much practice will be needed before the student will be able to make a satisfactory use of the Tables, and he will be required to take the subject in hand with great earnestness and determination. He is referred to the Treatise before mentioned, Snowball's Trigonometry, where he will find every requisite direction. A few easy examples are subjoined.

Ex. 1. Required the value of 3047.

5

Ex. 2. Let the value of 7/2/3 be required.

.. the value required is 1.07188 &c.

Ex. 3. Find a fourth proportional to the 6th power of 9, the 4th power of 7, and the 5th power of 5.

Let x be the required number; then

96: 7: 55: x, and x =

7' x 56
99

from which we may obtain the number of terms in any Geometric Progression, when the first term, common ratio, and sum, are given.

395. DEF. Interest is the consideration paid for the use of The rate of interest is the conmoney which belongs to another. sideration paid for the use of a certain sum for a certain time, as of

£1 for one year.

When the interest of the Principal alone, or sum lent, is taken, it is called Simple Interest; but if the interest, as soon as it becomes due, be added to the principal, and interest be charged upon the whole, it is called Compound Interest.

The Amount is the whole sum due at the end of any time, Interest and Principal together.

Discount is the abatement made for the payment of money before it becomes due.

* In this Ex. by ab" is meant a raised to the power expressed by b", and not a raised to the rth power.

396.

SIMPLE INTEREST.

To find the Amount of a given sum, in any time, at simple interest.

Let P be the principal, in pounds,

n the No. of years for which the interest is to be calculated*. r the interest of £1 for one year†,

M the amount.

Then, since the interest of a given sum, at a given rate, must be proportional to the time, 1 (year) : n (years) :: r: nr, the interest of £1 for n years; and the interest of P£ must be P times as great, or Pnr; therefore the amount M = P + Pnr.

397. From this simple equation, any three of the quantities P, n, r, M being given, the fourth may be found; thus

P =

Ex.

Or, which

What sum must be paid down to receive 600£, at the end of nine months, allowing 5 per cent. abatement ? is the same thing, what principal P will in nine months amount to 600£, allowing interest at the rate of 5 per cent. per annum?

398. To find the amount of a given sum in any time at compound interest.

Let R= 1 together with its interest for a year; then at the end of the first year, R becomes the principal, or sum due;

* When days, weeks, or months, not making an exact number of years, enter the calculation, n is fractional.

+ It must always be borne in mind that r is not the rate per cent. but only the hundredth part of it. Thus for 4 per cent. r = 0.04£, for 5 per cent. r = = 0·05£; and so on."

The amount at the end of the 2d year

= amount of R£ in 1 year

= RxR=R2,

= amount of R2 in 1 year

The amount at the end of the 3d

year

= R2× R = R3 ;

and so on; so that R" is the amount of 1£ in n years. And if P£ be the principal, the amount must be P times as great, or

COR. 2.

Ex. 1.

The interest M - P = PR" - P = P {R" − 1}.

=

What must be paid down to receive 600£ at the end of 3 years, allowing 5 per cent, per annum compound interest?

Ex. 2. Find the amount of 5£ in 2 years at 3 per cent., compound interest.

.. PR" = 5 × 1·0767 = 5.3835 = 5£. 7s. 8d., the amount required.

399. When compound interest is named, it is usually meant that interest is payable only at the end of each year; but there may be cases in which the interest is due half-yearly, quarterly, &c.; and then the amount found in the last Article will be altered. Thus, if r be the interest of 1£ paid at the end of a year, it has been shewn that the amount of P£ at the end of n years = P (1 + r)".

But if be the interest of 1£ paid at the end of each half-year, then

2