Question

Question #d4055

Answer 1

You need to clarify the question
To format use: hash symbol log_5(2x+1) hash symbol
This looks like: `Log_5(2x+1)`

Explanation:

Assumption: The question is:

`log_3(x)=-2` ......................................(1)
`log_5(2x+1)-log_5(x)=2`................(2)
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Pre amble:")`

Compare to `" "log_10(x)=3`
Another way of writing this is: `10^3=x`

Also `log_z(a) -log_z(b) = log_z(a/b)`

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Consider equation (1)")`

Write as `" "3^(-2)=x`

`color(blue)(=>x-1/3^2=1/9) ..........................(1_a)`
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Consider equation (2)")`

Write as:`" "log_5((2x+1)/2)=2`

`=>5^2=(2x+1)/2`

`=>25=x+1/2`

`color(blue)(=>x=24.5)..............................(2_a)`

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

`color(red)("Either I have interpreted the questions incorrectly or")`
`color(red)("the two equation are not connected!")`
`color(red)("It could be that the question is wrong!")`

`color(magenta)("Equation "(2_a) !=" Equation "(1_a)" "->" contradiction")`

Answer 2

A different interpretation of question

Explanation:

Assumption:

`log(3^x)=-2`................................(1)
`log(5^(2x+1))-log(5^x)=2`.... ..(2)
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Consider equation (1)

Write as `" "xlog(3)=2`

`color(blue)(=>x=2/log(3) ~~4.1918)` to 4 decimal places

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Consider equation (2)

Write as `" "(2x-1)log(5)-xLog(5)=2`

`=>log(5)(2x-1-x)=2`

`x-1=2/log(5)`

`color(blue)(x= 2/log(5)+1 ~~3.8613)` to 4 decimal places