CAIE A-Level Mathematics 3.2.1 Logarithm Laws

Use law of logarithm of a power

M1

$\log _{3}(2 x+1)=1+\log _{3}(x-1)^{2}$

Use $\log _{3} 3=1$

B1

$$\begin{aligned} & \log _{3}(2 x+1)=\log _{3} 3+2 \log _{3}(x-1) & {\left[\log 3\left(\frac{2 x+1}{(x-1)^{2}}\right)=\log _{3} 3 \text { or }\left(\frac{2 x+1}{(x-1)^{2}}\right)=3\right]} \end{aligned}$$

SC For candidates scoring M0 B0 due to combining logs before dealing with coefficient 2, and confusing coefficients, allow $\log _{3}(\ldots)=c$ leading to $(\ldots)=3^{c} \quad \mathbf{B1}$.

Obtain $3 x^{2}-8 x+2=0$ or $1.5 x^{2}-4 x+1=0$

A1

OE 3 terms only and =0 required.

Solve 3-term quadratic equation from part 3(a) or restart to find y

M1

$$\begin{aligned} & y=\frac{4 \pm \sqrt{10}}{6} \text { or } y=1.1937 \ldots \text { or } y=0.1396 \ldots & (x=2.3874 \text { or } x=0.2792) \end{aligned}$$

May solve for x but must find $y=\frac{x}{2}$ to gain M1.

Obtain answer 1.19

A1

CAO. 2 dp required.