对数练习题
Logarithms
Definition The logarithm of a number to a particular ba is the power (or index) to which that ba must be raid to obtain the number. This means that a logarithm is an index. Index or logarithm The number 8 written in index form is
8 = 23
ba
The equation can be rewritten in logarithm form as ba
log 2 8 = 3
Index or logarithm
The logarithm statement reads the logarithm of 8, to the ba 2 is 3 and is equivalent to the index statement 8 equals 2 to the power 3 or 2 to the power 3 equals 8. In general
ax = n
log an = x
(follow the arrows and read a x = n )
( a 0)
Examples
1. Change the following from index form to logarithm form. (a) 103 = 1000 using the expression above a = 10, x =3, n =1000 so
103 = 1000 log101000 = 3
(b)
2-5 =
1 32克隆好友
1 32
using the expression above a = 2, x = –5, n =
2-5 = 1 32 log 2 1 = 5 32
so
小肉牛Page 1 of 8
(c) 16 = 4
1 2
using the expression above a = 16, x =
1利润率的计算公式
1 2
, n =4
16 2 = 4
log16 4 =
1 2
2. Change the following from logarithm form to index form .
(a) log 2 16 = 4
the ba is 2,
so log 2 16 = 4
the number is 16, the index is 4
24 = 16 (follow the arrows and read 2 4 = 16 )
(b) log10 10 = 1
the ba is 10,
so
the number is 10, the index is 1
log10 10 = 1 101 = 10
1 (c) log 3 = 4 81
the ba is 3,
the number is
1 log 3 = 4 81
1 , the index is – 4 81 1 34 = 81
Note
1. log a 1 = 0 The logarithm of 1 to any ba is 0. 2 log a a = 1
The logarithm of a number to the same ba is 1.
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Exerci 1
Write in logarithm form (a) 32 = 9
1
段子大全爆笑(b) 10 4 = 10 000 (d) 102 = 0.01 (f) z 0 = 1
(c) 64 2 = 8 (e) x5 = y
Exerci 2
Write in index form (a) log10 100 = 2 (c) log10 0.001 = 3 (e) log 3 9 = x (b) log 2 32 = 5 (d) log 4 x = 3 (f) log a a = 1
Evaluating logarithms
To evaluate a logarithm: 1. write the number in index form, with same ba as the logarith
m 2. u the definition of a logarithm to evaluate.
Examples数学与应用数学专业就业方向
(1) Evaluate log 5 125 125 = 53 ∴ log 5 125 = 3 (2) Evaluate log10 0.0001 0.0001 = 10– 4 ∴ log10 0.0001 = 4
125 in index form with ba 5. The logarithm is 3.
equivalent logarithm statement
0.0001 in index form with ba 10. The logarithm is – 4. equivalent logarithm statement
Exerci 3
Without using a calculator, evaluate the following: (a) log 7 49 (c) log 2 128 (e) log 5 1 (g) log10 0.001 (b) log10 10 (d) log10 100 000 1 (f) log 3 27
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Solving equations (1) Examples
Equations of the type log a x = b and log x a = b can be solved for x by rewriting the equation in index form. Solve the following for x. (1) log 2 x = 5 log 2 x = 5 25 = x therefore x = 32 (2) log x 64 = 3 log x 64 = 3 x3 = 64
x 3 = 43 equate bas
therefore x = 4
Exerci 4
Solve the following equations for the unknown (a) log 3 x = 4 (b) log10 x = 3 1 (d) log y 36 = 2 (c) log a 4 = 2 1 2 (f) log x = (e) log x 27 = 3 9 3
Logarithm laws
The logarithm laws are obtained from the index laws and are: log a x + log a y = lo g a xy eg . log 2 5 + log 2 3 = log 2 (5 × 3)
= log 2 15 x y 12 3 = log 2 4
log a x log a y = log a
eg . log10 12 log10 3 = log10
log a x y = y log a x 1 = log a x x
eg . log 3 7 4 = 4 log 3 7 1 = log a 10 10
log a
eg . log a
log a 1 = 0 log a a = 1
eg . log10 1 = 0 eg . log 5 5 = 1
a log a x = x eg . 10log10 4 = 4 Note: It is not possible to have the logarithm of a negative number. Only logarithms with the same ba can be simplified using log laws.
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Examples
Express the following as a single logarithm. (Remember the logarithms must have the same ba if they are to be added or subtracted). (1) log 3 5 + log 3 20 log 3 10
U the laws for adding and subtracting logarithms.
log 3 5 + log 3 20 log 3 10 = log 3 ( 5 × 20 ) log 3 10 5 × 20 = log 3 10 = log 3 10
(2) log 2 5 + 3 log 2 3 2 log 2 6
3log 2 3 and 2 log 2 6 must be written as log 2 33 and log 2 62 before using addition
and subtraction laws.
月亮娃娃F 5× 3 I GH 6 JK F 15I = log G J H 4K
3
log 2 5 + 3log 2 3 2 log 2 6 = log 2 5 + log 2 33 log 2 62
2
= log 2
2
(3) Simplify and evaluate 1 log10 36 log10 15 + 2 log10 5 2 = log10 36 log10 15 + log10 52
1 2
write each term in the form log10 ( u laws for adding and subtracting logarithms
)
= log10 6 log10 15 + log10 25
6 × 25 15 = log10 10 = log10 =1
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Exerci 5
小人物的故事Express as a single logarithm and evaluate, if possible, without using a calculator . (a) log 4 8 + log 4 3 log 4 2 (b) log 2 5 + 2 log 2 4 + log 2
2 5
1 log10 50 log10 4 + 2 log10 3 2 (e) log a 4 + 2 log a 3 2 log a 6
