Form 2 Mathematics – LOGARITHMS

EXERCISE 1:

i). Write the following in standard form

17000
= 1.7 x 10⁴

ii) 0.00998
= 9.98 x 10^-3

iii). Write in standard form

0.000625
= 6.25 x 10^-4

8/300 correct to four significant figure

8/300 = 0.02666

Now 2.667 x 10^-2

2.667 x 10^-2

iv) If a = br and a = 8.4 x 10⁴ , b = 7.0 x 10² Find r.

solution:

a = 84 000

b = 700

Now

br = a

(700) (r) = 84000

r=120
r = 1.2 x 10²

iv) 3 = 9^1/2
log₃9 = 1⁄2

Example 3

If log₁₀0.01= y. Find y

Solution:

log₁₀0.01 = y

10^y = 0.01

10^y =1×10^-2

10^y=10⁰×10^-2

10^y=10^-2

y= -2

If log₁₀x=-3 find x

Solution:
log₁₀x = -3

10^-3 = x

x=0.001

EXERCISE 1
1. Write in standard form
i)405.06
ii) 0.912
Solution:
i) 405.06 = 4.0506 x 10²
ii) 0.912 = 9.12 x 10^-1

4. To solve for x
i) log₆x= 4
6⁴ = x
x = 1296

ii)x = log₃6561
3^x = 6561
x = 8

iii)log_x10= 1
x¹ = 10
x = 10

iv) log₄2 = x
4^x = 2
2^2x = 2¹
2x= 1
x = 1⁄2

BASE TEN LOGARITHM
– Is an logarithm of a number to base 10. Also known as common logarithm
example i) log₁₀5= log5
ii) log₁₀75 = log75

iii) log₁₀p = log p

SPECIAL CASES
(1). log_aa = x
a^x = a¹
x = 1

Generally log_aa = 1

Example
i) log₆6 = 1
ii) log10 = 1

(2) log_a(aⁿ) = x
a^x = aⁿ
x = n

Example i) log₄(4⁵) = 5
ii) log10^-3 = -3

Example 1

If log₅5 = log₂m Find m

Solution:

Given log₅25 + log₄x = 6, Find x

Solution:

EXERCISE 2.

Evaluate

i) log₂4096

ii) log0.0001

solution

i) log₂4096

let x = log₂4096
2^x = 4096

2^x= 2¹²

x = 12

∴log₂4096=12

ii) log0.0001

Solution:

Let x = log0.0001
10^x = 1/10000
10^x = 1/(10⁴)
10^x = 10^-4
x = -4

∴log0001=1

2) If log_k81 – log₂32= -1

Solution:
log_k81 – 5log₂2 = -1
log_k81 = -1 + 5
log_k81= 4
k⁴ = 81
k⁴ = 3⁴
k = 3

3. Given log₆y = log₇343. Find y

Solution:

log₆y = 3log₇7

log₆y = 3

6³ = y

216 = y

y = 216

4) Solve for m

i) log₈1 = m
8^m = 1 since aº=1 then

8^m=8⁰

m=0

ii) log₅m + log₃27 = 8
log₅m+ log₃3³= 8
log₅m +3 = 8
log₅m= 5
m= 5⁵

m = 3125

LAWS OF LOGARITHMS

MULTIPLICATION LAW

Suppose, log_ax = p and log_ay = q then
log_ax= p….(i)
log_ay= q….(ii)
Write equation (i) and (ii) into exponential form.
a^p = x………(iii)
a^q = y……..(iv)
Multiply equation (iii) and (iv)

xy = a^p x a^q
xy = a^{(p + q)} …….(v)

In equation (v) apply log_a both sides
log_a (xy) = log_aa^{(p + q)}
log_axy = (p + q) log_aa
log_axy= p + q
But p = log_ax
q = log_ay

Example

i) log₆(8 ×12) = log₈8 + log₆12
ii) log₄9 +log₄3 = log₄(9 ×3)

Example 1

i)Find x , If log₃x = log₃15 + log₃12

Solution:

log₃x = log₃15 + log₃12
log₃x = log₃( 15 ×12)
log₃x = log₃180
∴x = 180

Example 2

Given log₅20 = log₅4 + log₅x .Find x

Solution:

log₅20 = log₅4 + log₅x

log₅20 = log₅(4 × x)
log₅20 = log₅4x
∴20 = 4x

X = 5

Example 3

If log₈0.01= log₈(m ×2). Find m

solution

log₈0.01 = log₈( 2m)

∴ 0.01 = 2m

m = 0.01/2

m = 0.005

QUOTIENT LAW

Suppose, log_ax= p and
log_ay = q then
log_ax = p……..(i)
log_ay = q……..(ii)

Write equation (i) and (ii) into exponent form
a^p = x……(iii)
a^q = y…..(iv)

Divide equation (iii) and (iv)
x/y = a^p/a^q
x/y = a^{(p – q)} ….. (v)

In equation (v) apply log a both sides
log_a(x/y) = log_aa^(p-q)
log_a(x/y) = (p – q) log_aa
Butlog_aa = 1
log_a(x/y)= p – q ,where p= log_ax and q=log_ay

i) log₆( 8/12) = log₆8 – log₆12
ii) log₄9- log₄3 = log₄(9/3)

Example

If log₂20 = log₂x – log₂8.Find x

Solution:

log₂20 = log₂x – log₂8
log₂20 = log₂(x/8)
Now, 20 = x/8
X = 20 x 8
X = 160

Solution:

log₁₀25 – log₁₀9 + log₁₀360
log₁₀( (25 ×360 )/9)
=log₁₀1000
=log₁₀10³
=3log₁₀10
=3

2. If log_5a⁡x = log_5a⁡9 + log_5a⁡12. Find x

Solution:

log_5a⁡x = log_5a⁡9 + log_5⁡a12
log_5a⁡x= log_5a⁡( 9 ×12)
log_5a⁡x= log_5a⁡108
x = 108

3. If log_2a⁡5 = log_2ay + log_2a⁡0.001.Find Y

Solution:

i) log_2a⁡5 = log_2a⁡(y×0.001)

5⁡ = ⁡0.001y

y= 5/0.001

Y = 5000

ii)Find y if log₆⁡100 = log₆⁡5 + log_6⁡80 – log₆⁡y

Solution:

log₆⁡100 = log₆ (5 × 80)/ y

100= ⁡400/y

y = 4

4. If log a = 0.9031, log b = 1.0792 and log c = 0.6990. Find log⁡ ac⁄b

Solution

log ⁡ac⁄b =log_10⁡ a + log₁₀⁡c – log₁₀⁡b
=0.9031 +1.0792 -0.6990

∴ log ⁡ac⁄b = 1.2833

LOGARITHM OF POWER

If log_a⁡x = p then
X = a^p
Multiply by power in both sides xⁿ = a^np

Apply log _a both sides
log_axⁿ = log_aa^np
log_axⁿ = np

But p = log_ax

∴ log_axⁿ= nlog_ax

Example(1)

Evaluate

i) log₂ (128)⁶
ii) log₇ (343)⁸

Solution
i)log₂ (128)⁶ = 6log₂ 2⁷

= (7 x 6) log₂2

= 42 x 1

= 42

ii) Log₇ (343)⁸
Solution:
log₇ 343⁸ = 8log₇ 343
= 8log₇ 7³
= (8 x 3) log₇ 7
= 24

Example (2)

If log₅ 625^y = log₃ 729² .Find y.

Solution:
log₅ 625^y = log₃ 729²
log₅ 625^y= 2log₃ 729
ylog₅ 5⁴ = 2log₃ 3⁶
(y x 4) log₅ 5 = (2 x 6) log₃ 3
4y log₅ 5 = 12 log₃ 3
4y = 12

y=2/4

y = 3

LOGARITHM OF ROOTS