# How to algebraically solve 100ln(10)=xln(x) for x? **URL:** \ **Category:** Factual Questions\ **Created:** [October 16, 2014, 8:42pm UTC](https://boards.straightdope.com/t/how-to-algebraically-solve-100ln-10-xln-x-for-x/701529 "2014-10-16T20:42:44Z")\ **Posts on this page:** 10\ **Page:** 1
**Author:** ![R3d\_Anonymous](https://avatars.discourse-cdn.com/v4/letter/r/aeb1de/32.png) [@R3d\_Anonymous](https://boards.straightdope.com/u/R3d_Anonymous)\ **Post date:** [October 16, 2014, 8:42pm UTC](https://boards.straightdope.com/t/how-to-algebraically-solve-100ln-10-xln-x-for-x/701529/1 "2014-10-16T20:42:44Z")
My graphing calculator can’t display numbers higher than 10^100. In calculus, we were talking about finding the derivative of f(x)=x^x and I was curious to see what the graph looked like. I played around with it and little and basically noticed that after a certain x-value, if I try to evaluate the function, the calculator will say “OVERFLOW” (if the evaluation is greater than 10^10). Then I wanted to find out what the highest x-value is that I can plug in and still get an actual response from the calculator. Of course, I was able to through guess and check, find out that it was around 56.961… (if I remember correctly), but I wanted to see how one could algebraically solve this to get an exact answer. I set it up: 10^100=x^x ln(10^100)=ln(x^x) 100ln(10)=xln(x) There I am stuck (and I have tried rearranging it in all sorts of ways but none of them worked). Now I asked my teacher and he says the he doesn’t know of a way to solve that algebraically, but there might be one. Though he also says that not everything can be solved algebraically. Do any of _you_ know? This isn’t a homework problem. This is pure curiosity. ---
**Author:** ![Great\_Antibob](https://avatars.discourse-cdn.com/v4/letter/g/e47c2d/32.png) [@Great\_Antibob](https://boards.straightdope.com/u/Great_Antibob)\ **Post date:** [October 16, 2014, 8:52pm UTC](https://boards.straightdope.com/t/how-to-algebraically-solve-100ln-10-xln-x-for-x/701529/2 "2014-10-16T20:52:00Z")
This is a [transcendental equation](http://en.wikipedia.org/wiki/Transcendental_equation). There’s not going to be an algebraic solution here. Sometimes, transcendental equations may have closed form solutions, but that’s not common (and you don’t generally find such solutions algebraically). Usually, you need to use a numeric or graphical method to find solutions, as you have done. ---
**Author:** ![Reply](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/reply/32/15952_2.png) [@Reply](https://boards.straightdope.com/u/Reply)\ **Post date:** [October 16, 2014, 8:52pm UTC](https://boards.straightdope.com/t/how-to-algebraically-solve-100ln-10-xln-x-for-x/701529/3 "2014-10-16T20:52:15Z")
Wolfram can graph that: > **[100ln(10)=xln(x) - Wolfram|Alpha](https://www.wolframalpha.com/input/?i=100ln%2810%29%3Dxln%28x%29)** > > Wolfram|Alpha brings expert-level knowledge and capabilities to the broadest possible range of people—spanning all professions and education levels. ---
**Author:** ![Dr.Strangelove](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/dr.strangelove/32/6613_2.png) [@Dr.Strangelove](https://boards.straightdope.com/u/Dr.Strangelove)\ **Post date:** [October 16, 2014, 10:01pm UTC](https://boards.straightdope.com/t/how-to-algebraically-solve-100ln-10-xln-x-for-x/701529/4 "2014-10-16T22:01:31Z")
You can solve it via Newton-Raphson iteration if you like. We want to find the zero of: f(x) = x_ln(x) - 100_ln(10) f’(x) = ln(x) + 1 So we iterate: x’ = x - (x_ln(x) - 100_ln(10)) / (ln(x) + 1) Start with a guess of 50. x’ = 50 - (195.601 - 230.258) / (3.912 + 1) = 57.055 x’ = 57.055 - (230.734 - 230.258) / (4.044 + 1) = 56.961 (already at your answer) This method will roughly double your significant digits at each step, so with just one or two more iterations you’ll reach the limits of your calculator. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [October 17, 2014, 1:50am UTC](https://boards.straightdope.com/t/how-to-algebraically-solve-100ln-10-xln-x-for-x/701529/5 "2014-10-17T01:50:36Z")
Even when transcendental equations do have solutions, they’re generally of the form “Let’s create a new function that we’ll define as the solution to that particular problem”. Some problems come up repeatedly, so that those purpose-defined functions come in very useful, sometimes even to the point that we forget that they’re special functions. For instance, the problem f’’(x) = -f(x) shows up a lot, and so we invented the sine function to solve that problem. The problem you’re posing, though, is not one that comes up very often, so there might not be a suitable function for it, or if there is, it’s not well-known. ---
**Author:** ![Dr.Strangelove](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/dr.strangelove/32/6613_2.png) [@Dr.Strangelove](https://boards.straightdope.com/u/Dr.Strangelove)\ **Post date:** [October 17, 2014, 2:00am UTC](https://boards.straightdope.com/t/how-to-algebraically-solve-100ln-10-xln-x-for-x/701529/6 "2014-10-17T02:00:15Z")
> [@Chronos](#): > > The problem you’re posing, though, is not one that comes up very often, so there might not be a suitable function for it, or if there is, it’s not well-known. According to [Wolfram Alpha](http://www.wolframalpha.com/input/?i=zeroes+of+x*ln%28x%29+-+100*ln%2810%29), there is an answer… which uses the special-purpose [Lambert W function](http://en.wikipedia.org/wiki/Lambert_W_function). ---
**Author:** ![leahcim](https://avatars.discourse-cdn.com/v4/letter/l/b4bc9f/32.png) [@leahcim](https://boards.straightdope.com/u/leahcim)\ **Post date:** [October 17, 2014, 2:11am UTC](https://boards.straightdope.com/t/how-to-algebraically-solve-100ln-10-xln-x-for-x/701529/7 "2014-10-17T02:11:57Z")
Conversation I had many times in my career as a first-year calculus TA: **Me:** Solve exp(x) = x . **Them:** That’s easy – just take the logarithm of both sides… oh. ---
**Author:** ![am77494](https://avatars.discourse-cdn.com/v4/letter/a/d2c977/32.png) [@am77494](https://boards.straightdope.com/u/am77494)\ **Post date:** [October 17, 2014, 2:21pm UTC](https://boards.straightdope.com/t/how-to-algebraically-solve-100ln-10-xln-x-for-x/701529/8 "2014-10-17T14:21:33Z")
Its easy to do with a calculator (not an algebraic solution) but a easy numerical solution : Rearrange the equation x = 100\*ln(10)/ln(x) Assume x (old) = 100 Then x (new) = 100 \* ln(10)/ln(100) = 50 Plug it back in : x(new) = 100 \* ln(10)/ln(50) = 58.86 Back again and again (its easy to do in the calculator since you can use Ans as the variable:)Here’s what you will see : 56.50310354 57.07527014 56.93308392 56.96821833 56.95952438 56.96167494 56.96114292 56.96127453 56.96124198 56.96125003 ---
**Author:** ![R3d\_Anonymous](https://avatars.discourse-cdn.com/v4/letter/r/aeb1de/32.png) [@R3d\_Anonymous](https://boards.straightdope.com/u/R3d_Anonymous)\ **Post date:** [October 18, 2014, 2:26am UTC](https://boards.straightdope.com/t/how-to-algebraically-solve-100ln-10-xln-x-for-x/701529/9 "2014-10-18T02:26:07Z")
> [@Reply](#): > > Wolfram can graph that: > [100ln(10)=xln(x) - Wolfram|Alpha](http://m.wolframalpha.com/input/?i=100ln(10)%3Dxln(x)) It says it can be expressed like this in “exact form”: x = e^(W(100 log(10))) What’s that _W_? ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [October 18, 2014, 2:31am UTC](https://boards.straightdope.com/t/how-to-algebraically-solve-100ln-10-xln-x-for-x/701529/10 "2014-10-18T02:31:53Z")
> [@R3d\_Anonymous](#): > > It says it can be expressed like this in “exact form”: > > x = e^(W(100 log(10))) > > What’s that _W_? As posted: > [@Dr.Strangelove](#): > > According to [Wolfram Alpha](http://www.wolframalpha.com/input/?i=zeroes+of+x*ln%28x%29+-+100*ln%2810%29), there is an answer… which uses the special-purpose [Lambert W function](http://en.wikipedia.org/wiki/Lambert_W_function).