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Example 2. Given that the mantissa of the logarithm of 36741 is 5651510, we can at once write down the logarithm of any number whose digits are 36741. Thus

log 3674100 = 6·5651510,
log 36741

=4:5651510,
log 367.41 =2:5651510,
log •36741

= 7.5651510,

log:00036741=4.5651510, and so on.

207. In any set of tables of common logarithms the student will find the mantissa only corresponding to any set of digits.

It would obviously be superfluous to give the characteristic.

208. It is most important to remember to keep the mantissa always positive.

Example. Find the fifth root of .00065061.
Here

logo .00065061=4.8133207,
.: log10 (-00065061)* =3(4.8133207)=} (-4+ 8133207)

=f(-5+1.8133207)=-1+3626641=1.3626641, and

1-3626641=log 23050, .. the fifth root of .00065062=-23050 nearly.

EXAMPLES. LV.

(1) Write down the logarithms of 776:43, 7.7643, *00077643 and 776430. (The table gives opposite the numbers 77643, the figures 8901023.)

(2) Given that log10 59082=4.7714552, write down the logarithms of 5908200, 5.9082, 00059082, 590·82 and 5908·2.

(3) Find the fourth root of .0059082, having given that

log 5.9082=7714552; 4:4428638=log10 27724. (4) Find the product of .00059082 and ·027724, having given that •21431=log 16380 (cf. Question 3).

(5) Find the 10th root of •077643 (cf. Question 1), having given that •8890102=log 7:7448.

(6) Find the product of (-27724)2 and •077643. (See Questions 1 and 3; 7758288=log 59680.)

** 209. To transform a system of logarithms having a given base, to another system with a different base.

If we are given a list of logarithms calculated to a given base, we can deduce from it a list of logarithms calculated to any other base.

Let a be the given base ; let 6 be any other base.

Let m be any number. Then the logarithm of m to the base a is in the given list. Let this logarithm bel. Then

m=a?. We wish to find the logarithm of m to the base b. Let it be x. Then

m=b. But m=a?;

2

.: a = 6*; or, b=a*; 1

is the logarithm of 6 to the base a.

or,

Now the logarithm of 6 to the base a is given. For it is in the list of logarithms to the base a. 1

1
Thus,
= loga b; or, a =

;
log.b

loga m or,

logo m=

loga b

C

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Hence, to calculate the logarithins of a series of numbers to a new base b, we have only to divide each of the logarithms of the numbers to any given base a, by a certain divisor, viz. loga b.

If then a list of logarithms to some base e can be made, we can deduce from it a list of common logaritlıms, by

1 multiplying each logarithm in the given list by log, 10

Example. Show how to transform logarithms, having 5 for base to logarithms having 125 for base.

Suppose m=5, so that l=log, m.
Now 125=53, so that 3=log. 125,

2
and m=5*5=1253, so that = log125 m2.

3 Thus the logarithm of any number to base 5, divided by 3 (i.e. by log: 125), is the logarithm of the same number to the base 125. **210. The student will find that the logarithms of numbers cannot be calculated to the base 10 directly.

They are first calculated to the base 2:7182818, etc., which is the sum of the series 1 1

1
l+it
+

+etc. ad inf.
1.2

3 1.2.3.4
This number is called e.
And the constant divisor in this case is log. 10,

1

1 and log. 102:30258509

= '43429448, etc. When this constant divisor is transformed into a multiplier, this constant multiplier is called a modulus.

Example. Given that log10 12=1.0791812, shew how to transform common logarithms to logarithms having 12 for base. Here

101-0791812 = 12;

+

1.2.

1

12492662, etc.

.. 10=121*0791812 .. 10=12hx*02662.

* * EXAMPLES. LVI.

(1) Show how to transform logarithms having 2 for base to logarithms having 8 for base.

(2) Show how to transform logarithms having 9 for base to logarithms having 3 for base.

(3) Show how to transform common logarithms to logarithms having 2 for base (log10 2=3010300).

(4) Show how to transform logarithms having 3 for base to common logarithms (log 103=·4771213),

(5) Show how to transform common logarithms to logarithms having 3 for base.

(6) Given logio 2='3010300, find log, 10. (7) Given log10 7=8450980, find log, 10. (8) Given log10 2=3010300, find logg 10 and log32 10. * 211. We give here a formal proof of the following propositions : To prove

that (i) The logarithm of a product is equal to the sum of the logarithms of the factors.

(ii) The logarithm of a quotient is equal to the difference of the logarithms of the dividend and divisor

(iii) The logarithm of the power of a number is equal to the product of the logarithm of the number by the index denoting

the power.

(iv) The logarithm of the root of a number is equal to the result of dividing the logarithm of the number by the number denoting the root.

Let m and n be any two numbers.
Let h and k be their logarithms to any base a.
So that

loga m=h, loga n=k and

.: mra", n=al

Then (i) mxn=a* xat=a*+*, .. logam xn=h+k=loga m+loga no (ii) min=a* --at =qh-, : logamin=h- k=logam – loga ni (iii) m=(a")"=a, :: logama=rh=r loga m.

h I (iv) Vm=(am);=a*

=as, .- loga Vm= ==x loga m.

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* 212. The above is only a formal way of saying, that since logarithms are indices, therefore they obey the index law,

** MISCELLANEOUS EXAMPLES. LVII. (1) Find log, 8, log, 1, log, 2, log, 1, logo 128.

(2) Show that the logarithms of all except eight of the numbers from 1 to 30 inclusive, can be calculated in terms of log 2, log 3 and log 7.

(3) Show that the logarithms of the numbers 1 to 10 inclusive

may be found in terms of the logarithms of 8, 14, 21.

(4) The mantissa of the log of 85762 is 9332949. Find the log of V-0085762.

Find how many figures there are in the integral part of (85762)

(5) Find the product of 47-609, 476.09, 47609, *000047609, having given that log 4.7609=6776891 and •7107564=log 5.1375.

(6) What are the characteristics of the logarithm of 3742 to the bases 3, 6, 10 and 12 respectively.

(7) Having given that log 2=3010300, log 3='4771213 and log 7=•8450980, solve the following equations :

(i) 24 x 344 = 72, (ii) 324 =128 x 74-4,
(iii) 122=49,

(iv) 28=214-8.
(8) Given log10 7, find log, 490.
(9) Given log10 3, find log, 270.
(10) Given log10 2, find log, 10.

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