Solving Equations Using Logarithms

Edexcel IAL WMA12/01/P2/Jan 2021/Q3 (Logarithmic​​ Equation)

• Solve​​ 

7x+2 = 3 

giving your answer in the form​​ x =log7⁡a​​ where a is a rational number in its simplest form.​​ 

(3)

• Using the laws of logarithms, solve​​ 

1 +log2⁡y +log2⁡ y + 4=log2⁡5 - y

(5)

SOLUTION

i- Taking log on both sides of the equation. ​​ 

log7⁡7x+2=log7⁡3

Using the power rule to bring the power down as the coefficient.​​ 

x+2log7⁡7=log7⁡3

xlog7⁡7+2log7⁡7=log7⁡3

Remember,​​ loga⁡a=1; thus,​​ log7⁡7=1

x(1)=log7⁡3-2log7⁡7

We havent made​​ 2log7⁡7=2​​ so that when​​ 2log7⁡7​​ goes oon the RHS of the equation there the division can be applied. As the required answer has to be in the form of ​​ x =log7⁡a​​ as prescribed in the question.​​ 

x=log7⁡3-log7⁡72

x=log7⁡3-log7⁡49

Applying the division rule.

x=log7⁡3449 

ii- To solve the given expression, lets apply the multiplication and division rule on it.​​ 

log2⁡y+log2⁡y+4-log2⁡5-y=-1

log2⁡yy+45-y=-1

Remember,​​ loga⁡b=k​​ can be converted into exponential expression as​​ b=ak

yy+45-y=2-1

yy+45-y=12

2yy+4=5-y

2y2+8y=5-y

2y2+9y-5=0

Solving the quadratic equation by middle term breaking method.​​ 

2y-1y+5=0

y=12    y=-5

y​​ cannot be negative as the argument of log is neither zero nor negative. Hence, choosing the positive value of y

y=12