Solving Equations Using Logarithms

Edexcel IAL WMA12/01/P2/June 2019/Q8 (Exponentials Equations and Logarithms)

• Find the exact solution of the equation​​ 

82x + 1= 6 

giving your answer in the form​​ a + blog2⁡3,​​ where a and b are constants to be found.​​ 

(4)​​ 

• Using the laws of logarithms, solve​​ 

log5⁡7 – 2y= 2log5⁡ y + 1 – 1 

(5)

SOLUTION

i-​​ Usng the laws of logarithm.

log2⁡82x+1=log2⁡6

Applying the power rule to bring (2x+1) down as the coefficient.

2x+1log2⁡8=log2⁡2 x 3

Applying the multiplication rule on the RHS of equation.​​ 

2x+1log2⁡23=log2⁡2+log2⁡3

2x+1×3×log2⁡2=log2⁡2+log2⁡3

Remember,​​ loga⁡a=1; thus,​​ log2⁡2=1

2x+1×3×1=1+log2⁡3

6x+3=1+log2⁡3

6x= -2+log2⁡3

x=-13+16log2⁡3

ii-​​ Using the laws of logarithm.

1=2log5⁡y+1=log5⁡7-2y

Applying the power rule in reverse that is changing the coefficient (2) as the power of (y+1).  ​​​​ 

1=log5⁡ y+12-log5⁡ 7-2y

Applying the division rule,​​ 

1=log5⁡y+127-2y

Writing this exponential equation into log equation. Thereofre, the base in of the indices becomes the base of log. (Remember,​​ loga⁡b=k​​ can be converted into exponential expression as​​ b=ak).

51=y+127-2y

35-10y=y+12

35-10y=y2+2y+1

y2+12y-34=0

y2+12y=34

Using completing square method to solve the quadratic equation.

y+62-36=34

y+62=70

y+6= ±70

y=-6±70