Question

How do you solve `log x + log (x-3) = 1`?

Answer 1

I found `x=5` if the base of the log is `10`.

Explanation:

We can take advantage of the property of logs that says:
`log(x)+log(y)=log(xy)`
to get in your case:
`log[x(x-3)]=1`
Now the problem is the base of your log...
`color(red)(->)` If the base is `10` we write:
`log_(10)[x(x-3)]=1`
apply the definition of log:
`x(x-3)=10^1`
solve for `x`:
`x^2-3x-10=0`
apply the Quadratic Formula:
`x_(1,2)=(3+-sqrt(9+40))/2=(3+-7)/2`
two solutions:
`x_1=(3+7)/2=5`
`x_2=(3-7)/2=-2` NOT because you'd get a negative argument of the log in the original expression.

If the base is NOT `10` use your specific base at the above stage marked:
`color(red)(->)`