LOGARITHMIC EQUATIONS

Any mathematic questions with equal to sign (=) is called equation

logarithm equation are algebraic equation involving logarithms.

      WORKED EXAMPLES



Example 1:


Solve the equation log₃x² = -2

Solution:

3⁻² = x² ⇒ from definition

Add square root to both side √(3⁻²) = √(x)²

3⁻² ˣ ½ = x² ˣ ½

3⁻¹ = x

1/3 = x




Example 2:


Calculate the value of log₂ᵃ if logₐ4 =1/2


Solution


a^½ = 4

a^½ˣ² = 4²

a = 16


Hence,

log₂a = log₂16 = log₂2⁴  = 4log₂2 = 4 x 1
= 4


Example 3:

log₈32 = y + 2

8⁽ʸ ⁺²⁾ = 32

2³⁽ʸ ⁺² ⁾=2⁵

3(y + 2) =5

3y + 6 =5

3y = 5 -6
y =-1/3



Example 4:

Find x if log₃2 + log₃⁽²ˣ ⁺ ³⁾ - log₃⁽ˣ ⁻¹⁾ = 2 


Solution:

log₃ (2) x (2x + 3)÷( x - 1) = 32

log₃ (2) x ( 2x +3) ÷ ( x - 1) =log₃3 x 2

(2)(2x +3)/(x -1) = 3 x 2

(4x + 6)/ (x - 1) = 6

6( x - 1) = (4x + 6)

6x - 6 = 4x + 6

6x - 4x = 6 + 6

2x = 12

x = 12/2

 = 6


Example 5:

log₄( a² + a + 10) - log₄(a² + a - 5) = 1/2, what are the values of a 


Solution:

log₄(a² + a + 10)/(a² + a - 5) = 1/2

log₄ ( a² + a + 10)/ (a² + a - 5) =log₄4 (1/2)

a² + a + 10/( a² + a - 5) =4(1/2)

a² + a + 10/ ( a² + a - 5) = 2

a² + a + 10= 2(a² + a  - 5) 

a² + a + 10 = 2a² + 2a - 10

Collect like term

a² + a - 20=0

a² +5a -4a - 20 =0 ⇒ quadractic equation

a(a + 5) -4(a + 5) =0

(a - 4)( a + 5) =0

a = 4 or a = -5


hence, the values are 4 and -5



Example 6:

₄log₄3y - 2 =10

3y - 2 =10

3y = 10 + 2

3y = 12

y = 4



Example 9:

log2 = z, express log12.5 in term of z

log125/10 = log25/2
log(25x4)/(2 x 4) =

log 100/ 8

log10²/ 2³

log10² - log2³

2log10 - 3log2
(log10 =1)
2 x1 -3 x x
2 - 3x




Example 9:

Given that log₂a = log₈4. Find a


Solution:

log₂a = log₈4

log₂a = log4/log8

log₂a= log2²/log2³

log₂a = 2/3

2^2/3 = a

( ^ means raise to √ power)

(³√2)² 


See...FINDING LOGARITHMS WITHOUT USING TABLES