Solving Equations Using Logarithms

Edexcel IAL WMA12/01/P2/Oct 2022/Q4 (Application of Laws of Logarithm)

The weight of a baby mammal is monitored over a 16‑month period.​​ 

The weight of the mammal, wkg, is given by

w =loga⁡t + 5 –loga⁡4             2≤ t≤ 18 

where t is the age of the mammal in months and a is a constant.​​ 

Given that the weight of the mammal was 10kg when t = 3​​ 

• show that a = 1.072 correct to 3 decimal places.​​ 

(3)​​ 

Using a = 1.072​​ 

• find an equation for t in terms of w​​ 

(3)​​ 

• find the value of t when w = 15, giving your answer to 3 significant figures.​​ 

(2)

SOLUTION​​ 

a- It is given that the weight of the mammal is 10 kg when t is 3. Substituting the values in the given equation, to find the value of​​ a.​​ 

w=loga⁡t+5-loga⁡4

10=loga⁡3+5-loga⁡4

10=loga⁡8-loga⁡4

Now, applying the laws of logarithms.​​ 

Applying the division rule to solve the equation further.

10=loga⁡84 

10=loga⁡2

a10=2

a= 210

a=1.07177…

a=1.072

b- ​​ When​​ a=1.072, the equation of t in terms of w is given as follow

w=loga⁡t+54 

Rewriting the logarithm expression using an exponent.​​ 

aw=t+54

4aw-5=t

t=41.072w-5

c- Finding the value of t, when w is 15 using the expression derived in the previous part.​​ 

t=41.072w-5

t=4 1.07215-5

t=6.35