A Nice Homemade Exponential Log Equation

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x^{1-log(x)}=1/100
#ChallengingMathProblems #ExponentialEquations #Logarithms
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[RazorM]

I just did the 2nd method on my own without any help. Syber math you taught me a lot of math. Your channel is brilliantly simple and the homemade equations are fun to solve. An easy method to teach complex problem solving, really. Great stuff as always!

[scottleung]

I used the first method, but with substitution.

[popitripodi]

I used the first method ❤❤nice and easy

[bobbyheffley]

1/100 is a power of 10, so common logarithms simplify the work involved.

[roberttelarket]

You'll probably need a negative exponent.
x = 10 won't work.
X = 100: 100^(1 - 2) works.

[zawatsky]

Только сейчас подумал, что одно и то же уравнение, но при разных корнях, может сводиться и к x², и к 1/x. Это выглядит странно, ведь обычно все выражения упрощаются одинаково...

[goldfing]

x^(1 - lg(x)) = 1/100
Take the lg() on both sides:
lg(x^(1 - lg(x)) = lg(10^(-2))
Simplify:
(1 - lg(x)) * lg(x) = -2
Substitute t = lg(x) resp. x = 10^t :
(1 - t)*t = -2
t - t^2 = -2
t^2 - t - 2 = 0
Multiply by 4:
4t^2 - 4t - 8 = 0
Add 8:
4t^2 - 4t = 8
Add 1:
4t^2 - 4t + 1 = 9
Pe3fect squares on both sides:
(2t - 1)^2 = 3^2
Take the square root:
2t - 1 = +-3
2t = 1 +- 3
t = (1 +- 3)/2
t1 = 4/2 = 2
t2 = -2/2 = -1
Since t = 10^x > 0, only r1 makes sense. We get
x = 10^2 = 100
as the only solution. Proof:
100^(1 - lg(100)) = 100^(1 - 2) = 100^(-1) = 1/100
Correct.

[carlyet]

before: I get x =1/10 now i watch to see what I missed.

[walterwen]

A Nice Homemade Exponential Log Equation: x^(1 – logx) = 1/100; x = ?
log[x^(1 – logx)] = log(1/100), (1 – logx)logx = logx – (logx)^2 = – 2
(logx)^2 – lgx – 2 = (logx + 1)(logx – 2) = 0, logx + 1 = 0 or logx – 2 = 0
logx = – 1 = log(1/10); x = 1/10 or logx = 2 = log(10^2); x = 10^2 = 100
Answer check:
x = 1/10, x^(1 – logx) = (1/10)^(1 + 1) = (1/10)^2 = 1/100; Confirmed
x = 100, 100^(1 – log100) = (100)^(1 – 2) = 1/100; Confirmed
Final answer:
x = 1/10 or x = 100

[belleringr]

‘cause the beauty of Mathematics is the solutions are infinity..!
(1 or 2 maybe 3)😉👍 what can you see and perceive —>>🤍∞👀