Solving equations with logarithms I'm having trouble with solving equations that has logarithms in

rigliztetbf

rigliztetbf

Answered question

2022-06-07

Solving equations with logarithms
I'm having trouble with solving equations that has logarithms in them. For example:
x log ( x ) = 100 x
How can I solve this? I have reed about how to do it but when I try to do the same with this example I just got stuck. The only thing I can think of doing is to draw the lines and see where they cross but I only get one answer x = 10 but there should be two and I will need to show how I came up with the answer by showing calculations. Please help.

Answer & Explanation

trajeronls

trajeronls

Beginner2022-06-08Added 21 answers

So you have x lg x = 100 x . First of all, you have logarithms AND you have an exponential function. This power functions are ugly and you may remember about the special property of logarithm:
log a b c = c log a b (there should be modulo, btw).
So you can apply a logarithm to both sides. What base — well, you already have a logarithm to base 10 AND you have 100 in the right side. So, lg is a good idea, indeed.
Now you have lg x lg x = lg 100 lg x
That means lg 2 x + lg x 2 = 0 — and you have quadratic equation.
The only thing now is to solve it and be careful.
excluderho

excluderho

Beginner2022-06-09Added 8 answers

The key is to take the log of both sides and then solve your equation:
log x log x = log 100 x ( log x ) ( log x ) = log 100 log x ( log x ) 2 + log x log 100 = 0 log x = 1 ± 1 + 8 log 10 2 x = e 1 ± 1 + 8 log 10 2

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