Machinery's Handbook, 31st Edition
38 LOGARITHMS Logarithms in any base can be converted to another base by the formula: log b = Logarithms of bases other than 10 and e were often converted to these bases, since the values were easy to look up on the common or natural log tables. The formula also can convert between the two most-often used bases: Base to base 10: so e x log log a a x x b x x log . ≈ x ln ln = ≈ 0 4343
x e
log log
log .
04343
x
ln
lnx 2.3026
e
x
x
x
=
Base 10 to base
:
log
so
ln
2.3026
log
≈
≈
ln
10
Example : Convert ln 4 to log 4 using the conversion, log . x ≈ 0 4343 log 4 ≈ 0.4343 ln 4 ≈ (0.4343) (1.3863) ≈ 0.60207 Example : Express log 2 9 in terms of the natural logarithm: log . 3 170 = = ≈ e
:
x
ln
log log
9 2
ln ln
9 2
2 9
e Using Calculators to Solve Logarithms.—To find a common logarithm on a scientific cal - culator (hand-held or online), the log key is used. To find a natural logarithm the ln key is used. Depending on the calculator used, either the number is entered before the log key is pressed, or log is entered before the number. The correct sequence can be seen if an error message results. For example, log 6 is found by this sequence on the typical scientific calculator: Example : To find log 6, press in sequence, 6 log to get display 0.77815125038 . . . To find the common antilog of a given number, the 10 x key is used. The find the natural antilog, the e x key is used. This kind of problem is often asked as ln x = 4, meaning e 4 = x , so one must know the definition of log b x = y to get the answer by calculator. Example : To find x in ln x = 4, press in sequence, 4 e x get the display 54.5980015003 . . . On calculators without the 10 x and e x keys, the x y key enables, substituting 10 or e (2.718281 . . .) for x and the logarithm of the number sought for y . On some calculators, while the log and ln keys are used to find common and natural logarithms, the same keys in combination with the INV , or inverse, key are used to find the number corresponding to a given logarithm. Solving an Equation Using Logarithms.—Solving exponential and logarithmic equa- tions is possible because of the following properties of logs and exponents, which are true for any base: If . If , then x y x y x y a a x y = = = = log log for a > 0, a ≠ 1. , and x , y > 0, then Both statements are true in the other direction, too: If , then then log log . . x y x y a a x y x y = = = = If Example 1: Find the square root of 754. Solution x x : . log log log log / Let Then = = = = ≈ 754 754 754 754 2 1 2 . . . 14387 ≈
1 2
8774 2
. 14387
log . ≈ x
x
=
So,
, 14387
hence,
10
. 27 460 .
That is, 754 27 460 ≈ . .
≈
Example 2: Solve 4 x = 7 x – 3 for x . Solution x x 4 7 3 = − : l
→ apply property
→
x x og log 4 7 =
−
3
x
x = −
log ( 4
3
)(l
og ) 7
→
→
x
log log log 4 7 3 7 = − x
= 0 6021 0 8451 3 0 8451 − . . ( . ) x x
distribute on the right
by calculator
→
→
0 8451 0 6021 2 5353 0 243 . . . . x x x − =
x ≈ , so
=
10433 .
3 08451 ( . )
proceed with algebra
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