Question

Solve the following logarithmic equations
log(10x + 5) − 3 = log(x − 5)

Answer 1

`x=(-5*(1+e^(3)) )/(10-e^(3) )`

Explanation:

Rewrite the equation, adding 3 to both sides and subtracting log(x-5) from both sides:

`log(10x+5)-log(x-5)=3`

Using the next propierty:

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

`log(10x+5)+log((1)/(x-5))=3`

Using this propierty:

`log(x*y)=log(x)+log(y)`

`log((10x+5)/(x-5))=3`

Cancel logarithms by taking exp of both sides:

`e^{log((10x+5)/(x-5))} =e^(3) \\(10x+5)/(x-5)=e^(3)`

Multiplying both sides by x-5 and factoring:

`x(10-e^(3) )=-5-5e^(3)`

Solving for x multiplying both sides by
`10-e^(3)`

`x=(-5-5e^(3) )/(10-e^(3) ) =(-5*(1+e^(3)) )/(10-e^(3) )`