Question

Solve the equation without using a calculator


`\bf\log_x\big(2+√(3) \big)=log_(\big(x+6\big))\big(1+2√(3) \big).`

Answer 1

• 
  `x=7+4√(3)`

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Given

• 
  `log_x(2+√(3))=log_(x+6)(1+2√(3))`

Solve it in below steps

• 
  `2log_x(2+√(3))=2log_(x+6)(1+2√(3))`
• 
  `log_x(2+√(3))^2=log_(x+6)(1+2√(3))^2`
• 
  `log_x(7+4√(3))=log_(x+6)(13+4√(3))`
• 
  `log_x(7+4√(3))=log_(x+6)(7+4√(3)+6)`

Let 7 + 4√3 = t, then

• 
  `log_xt = log_(x+6)(t+6)`
• 
  `logt/logx=log(t+6)/log(x+6)`
• 
  `logx/log(x+6)=logt/log(t+6)`
• 
  `log_(x+6)x=log_(t+6)t`
• 
  `x = t`

The value of x is

• 
  `x=7+4√(3)`