C3 help please?

Hi im doing the review exercises for C3 in my text book, and the question is 'show that the equation x=ln(x+5) has a root between x=1 and x=2'. i just cant seem to do it :/ help would be much appreciated!

Hi im doing the review exercises for C3 in my text book, and the question is 'show that the equation x=ln(x+5) has a root between x=1 and x=2'. i just cant seem to do it :/ help would be much appreciated!

What have you tried?

ok ive tried getting rid of the ln getting x+5=e^x? i suppose you need to make x the subject now right?

Hint:

Let

$f(x)= x-\ln(x+5)$

and notice that if, at x=a, f(x) crosses the x axis, it must be above on one side of a and below on the other.

so are you saying to use trial and error?

You are not trying to find x

RTQ

so are you saying to use trial and error?

Well, no.

$\displaystyle x=\ln(x+5) \Rightarrow x-\ln(x+5)=0$

Thus if you let f(x) equal the LHS, you need to show that there is a root of f(x)=0 between 1 and 2 (i.e it crosses the x axis)

Use the fact that, to cross the x-axis, it must be above the x axis at one point (on one side of the root), and below on the other.

See if you can relate what I'm saying to the values of f(x) when you substitute 1 and 2 in