Solving Equations Using Logarithms

Edexcel IAL WMA12/01/P2/June 2022/Q4 (Logarithm,​​ Simultaneous Equation)

In this question you must show all stages of your working.

Give your answers in fully simplified surd form.

Given that a and b are positive constants, solve the simultaneous equations​​ 

a - b = 8

log4⁡A +log4⁡b=3 

(6)

SOLUTION​​ 

Lets start with converting the equation 2 from logarithm into expenential form.​​ 

Applying the multiplication rule here.

log4⁡ab=3

Now, applying the power rule, to bring the power down as the coefficient.

43=ab

ab=64

Now, substituting the value of a from equation 1 into the above derived equation.​​ 

a-b=8  -1 

a=b+8

So,​​ ab=64​​ becomes​​ 

b+8b=64

b2+8b-64=0

b2+8b=64

Solving the quadratic equation by using completing square method.​​ 

b+42-16=64

b+42=80

b+4=±80

b+4=±45

b=-4±45 

We will be considering only positive value of​​ b​​ (b= -4+45) because we know that the argument of logarithm can neither ​​ be negative nor be zero. ​​ So rejecting​​ b= -4-45

Now, substituting the value of b in equation 1.​​ 

a-b=8  -1 

a=b+8

a=-4+45+8

a=4+45

Hence,​​ a=4+45​​ &​​ b= -4+45.