Thread: Logarithmic inequality

Solve for x.

Is there an algebraic solution for this? I plugged in values for , giving me what I suspect is right, x < 1. But how to be sure?

Originally Posted by rowe

Solve for x.

Is there an algebraic solution for this? I plugged in values for , giving me what I suspect is right, x < 1. But how to be sure?

1. Graph the function and check for which values of x the values of f(x) are negative.

2. attempt: Solve
for x. Since are monotically increasing there exists only one solution. By first inspection x = 1.
Therefore the inequality is true for all x < 1.

Originally Posted by rowe

Solve for x.

Is there an algebraic solution for this? I plugged in values for , giving me what I suspect is right, x < 1. But how to be sure?

Alternatively,





is a line decreasing at passing through
the y-axis at (0,4) and the x-axis at (4,0).

is an increasing function passing through the y axis at (0,1), so it cuts f(x) between x=0 and x=4.

When x=1, both f(x) and g(x) are 3.

Originally Posted by earboth

1. Graph the function and check for which values of x the values of f(x) are negative.

2. attempt: Solve
for x. Since are monotically increasing there exists only one solution. By first inspection x = 1.
Therefore the inequality is true for all x < 1.

@earboth

1. it would be much easier if he draw 3^x and 4-x and find values where 4-x is "bigger" then 3^x


2. I do not what made you think that if they're "monotically increasing there exists only one solution"?

example: x^3+5 and 3^x, both monotonically increasing but have 2 intersections.

Hello, rowe!

Solve: .

There is no algebraic solution, but there is a graphical solution.

We have: .

The question becomes: when is below ?


The graphs look like this:

Code:

              |
            * |     *
              *
              | *  *
              |   o 
              | * : *
              *   :   *
          *   |   :     *
    *         |   :       *
  - - - - - - + - + - - - - * - -
              |   1           *
              |

As you pointed out, the curves intersect at

You answer is correct: .