1. Find `log(10,2x)=1`
Solution:
`log(10,2x)=1`
Now, `log_(10)(2x)=1`
`=>log_(10)(2x)=1`
`=>2x=10^1`
`=>2x=10`
`=>x=10/2`
`=>x=5`
2. Find `log(98+sqrt(x^2-12x+36))=2`
Solution:
`log(98+sqrt(x^2-12x+36))=2`
Now, `log(98+sqrt(x^2-12x+36))=2`
`=>log(98+sqrt(x^2-12x+36))=2`
`=>98+sqrt(x^2-12x+36)=10^2`
`=>98+sqrt(x^2-12x+36)=100`
`=>sqrt(x^2-12x+36)+98=100`
`=>sqrt(x^2-12x+36)=100-98`
`=>sqrt(x^2-12x+36)=2`
`=>x^2-12x+36=4`
`=>x^2-12x+32=0`
`=>x^2-4x-8x+32 = 0`
`=>x(x-4)-8(x-4) = 0`
`=>(x-4)(x-8) = 0`
`=>(x-4) = 0" or "(x-8) = 0`
`=>x = 4" or "x = 8`
`x=4,x=8`
3. Find `log(x+1)+log(x-1)=log3`
Solution:
`log(x+1)+log(x-1)=log(3)`
Simplify LHS `=log(x+1)+log(x-1)`
`=log(x+1)+log(x-1)`
`=log((x+1) xx (x-1))`
`=log(x^2-1)`
Now, `log(x^2-1)=log(3)`
`=>log(x^2-1)=log(3)`
`=>x^2-1=3`
`=>x^2=3+1`
`=>x^2=4`
`=>x=2`
4. Find `log(2,x+1)=log(3,27)`
Solution:
`log(2,x+1)=log(3,27)`
Simplify RHS `=log(3,27)`
`=(log(27))/(log(3))`
`=(log(3^(3)))/(log(3))`
`=(3log(3))/(log(3))`
`=3`
Now, `log_(2)(x+1)=3`
`=>log_(2)(x+1)=3`
`=>x+1=2^3`
`=>x+1=8`
`=>x=8-1`
`=>x=7`
5. Find `log(10,log(10,x))=0`
Solution:
`log(10,log(10,x))=0`
Simplify LHS `=log(10,log(10,x))`
`log(10,x)=log_(10)(x)`
`log(10,x)`
`=log_(10)(log_(10)(x))`
Now, `log_(10)(log_(10)(x))=0`
`=>log_(10)(log_(10)(x))=0`
`=>log_(10)(x)=10^0`
`=>log_(10)(x)=1`
`=>x=10^1`
`=>x=10`
This material is intended as a summary. Use your textbook for detail explanation.
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