Question

Solve the equation for x, accurate to three decimal places:logx + log(x − 2) − log(x + 4) = 0.x = −1x = 4x = −1 and x = 4x = 0.602

Answer 1

Given:

`logx+log(x-2)-log(x+4)=0`

Applying the laws of logarithm,

`\begin{gathered} logM+logN=logMN \\ logM-logN=log((M)/(N)) \\ If\text{ }log_ab=x,\text{ then }a^x=b \end{gathered}`

Hence, the equation becomes;

`\begin{gathered} logx+log(x-2)-log(x+4)=0 \\ \\ Applying\text{ the laws of logarithm,} \\ \\ logM+logN=logMN \\ logM-logN=log((M)/(N)) \\ \\ Then, \\ \\ log((x(x-2)))/(x+4)=0 \\ log((x^2-2x))/(x+4)=0 \\ \\ Also,\text{ applying the law of logarithm} \\ If\text{ }log_ab=x,\text{ then }a^x=b. \\ \\ Hence, \\ log_(10)((x^(2)-2x))/(x+4)=0 \\ 10^0=(x^2-2x)/(x+4) \\ 1=(x^2-2x)/(x+4) \\ Cross\text{ multiplying,} \\ x^2-2x=x+4 \\ \\ Collecting\text{ all the terms to the left-hand side to form a quadratic equation;} \\ x^2-2x-x-4=0 \\ x^2-3x-4=0 \end{gathered}`

Solving the quadratic equation,

`\begin{gathered} x^2-3x-4=0 \\ x^2+x-4x-4=0 \\ x(x+1)-4(x+1)=0 \\ (x-4)(x+1)=0 \\ x-4=0,\text{ }x+1=0\text{ } \\ x=0+4,\text{ }x=0-1 \\ x=4,\text{ }x=-1 \end{gathered}`

Therefore, the correct answer is x = 4