Math Is Fun Forum

tony123

Member

Registered: 2007-08-03

Posts: 229

Find the real solutions

Find the real solutions of the  two equations

(1)

(2)

Bob

Administrator

Registered: 2010-06-20

Posts: 10,621

Re: Find the real solutions

hi tony123,

I think these are valid:

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

tony123

Member

Registered: 2007-08-03

Posts: 229

Re: Find the real solutions

There are some mistakes in your solution dear Bob

Bob

Administrator

Registered: 2010-06-20

Posts: 10,621

Re: Find the real solutions

With one or both?

Please give more details.

LATER EDIT:

Ok I see what is wrong (I think) with each one, so it's back to the drawing board for me.  This may take a while.

Bob

Last edited by Bob (2022-01-12 01:41:15)

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

Bob

Administrator

Registered: 2010-06-20

Posts: 10,621

Re: Find the real solutions

Hurray! I've got an answer for (2) that is confirmed by wolfram alpha.  Very busy today so I'll post it when I've got more time.

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

Bob

Administrator

Registered: 2010-06-20

Posts: 10,621

Re: Find the real solutions

Here we go:

All real log functions are undefined for x  ≤ 0 so x > 11 or the log powers don't work.

Replace x by X = x - 11

The equation becomes

For brevity I will use all logs in base 2 from now on.

case 1. log(X) = 0

case 2. if log(X)  ≠ 0 then it can be cancelled leaving

Raise 2 to the power of each side (you could also call this antilog}

The quadratic formula has one negative answer which contradicts the domain for X so

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

Bob

Administrator

Registered: 2010-06-20

Posts: 10,621

Re: Find the real solutions

And now for Q1:

First some preliminaries:

If

... A

and if

... B

And now the question

All logs in base 7

using A becomes

Divide by x

Replace 1 by log(7) and simplify the logs

Using A again gives

I will replace log(x) with X

Now what does the graph of the left hand side function look like?

It has two components. The first is a standard power graph, going through (0,1) and rising thereafter.  The second is is similar but, as 3/7 is under 1, it will come down from large y values to the left of the y axis, again go through (0,1) and then drop off to zero as X gets larger.

Together the second component will dominate when X is negative; then the first will dominate.  So the graph will start high, drop down to (0,2) and at some stage rise again.  Note that (0,2) gives one solution to the question.

Differentiating using B

The ln(3/7) is negative so this will give one turning point. It must be a minimum because of the overall shape already established; (0,2) could be the minimum; otherwise there will be exactly one more solution.

I can see that X = 1 gives x = 7 leading to 11/7 + 3/7 = 14/7 = 2. So here is the second solution.

So the equation has two solutions, x = 1 and x = 7.

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

tony123

Member

Registered: 2007-08-03

Posts: 229

Re: Find the real solutions

Now we can say great job
Thanks bob

pamshaw

Member

Registered: 2021-12-07

Posts: 21

Re: Find the real solutions

The problem statement here is a mathematical equation for which we have to find the answer for two equations given in the statement. The statement is basically logarithm equations that are integrated into the arithmetic situation.
Discussion on the problem highlights some important points of the equation, and the solution solved gives a few pointers as well to the community. They found some issues with the solution and highlighted them. The discussion also points out that the community has found some steps that can improve the results of the equation.
The community provides the solution of the equation by solving the equation step by step, helping the little bright minds to understand the steps and get up to speed with brilliant minds. The equation solution is very well explained for those who are not bright and sharp to understand things on the go and for those who can make sense of things on the  go.
With all the explanation, the equations solutions, the graphical presentations, the real solution of the equations have been provided, and all the necessary means and explanations have already been pointed out here,
X = \frac{-1}{2} + \frac{\sqrt{5}}{2} \implies x = 11.618...