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Math Olympiad Logarithmic Equation | Exponent Simplification #exponent#olympiad #simplification#exam
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Комментарии : 40

@walterwen2975 20 дней назад

Math Olympiad Logarithmic Equation: (x - 5)^[log(5x - 25)] = 2; x = ? (x - 5)^[log(5x - 25)] = (x - 5)^log[5(x - 5)] = (x - 5)^[log5 + log(x - 5)] log(x - 5)^[log5 + log(x - 5)] = [log5 + log(x - 5)]log(x - 5) = log2 [log(x - 5)]² + log5[log(x - 5)] - log2 = 0 Let: y = log(x - 5), y² + (log5)y - log2 = y² + [log(10/2)]y - log2 = 0 y² + (log10 - log2)y - log2 = (y - log2)(y + log10) = 0 y - log2 = 0, y = log2 or y + log10 = 0, y = - log10 = log(1/10) y = log(x - 5) = log2, x - 5 = 2; x = 7 or x - 5 = 1/10; x = 5 + 1/10 = 5.1 Answer check: x = 7: (x - 5)^[log(5x - 25)] = 2^(log10) = 2¹ = 2; Confirmed x = 5.1: (0.1)^(log0.5) = 2; Confirmed The calculation was achieved on a smartphone with a standard calculator app Final answer: x = 7 or x = 5.1

@kassuskassus6263 15 дней назад

Nice job !

@superacademy247 15 дней назад

Thanks!

@UKPEINDANIELU. 20 дней назад

Fantastic

@superacademy247 20 дней назад

Thank you so much 😀

@user-tb7nd2ut9t 5 дней назад

Nice job ❤

@superacademy247 5 дней назад

Thanks 😆 I'm glad you like it 👍

@hossamwalid3599 5 дней назад

Amazing

@superacademy247 4 дня назад

Thanks!

@carlosharmes2378 8 дней назад

Nice 2 & 5 equation... 💌

@superacademy247 8 дней назад

Keep watching

@rfnobando 6 дней назад

When you have this: log(5m) * log(m) = log(2) you can do this, 'cause log(10) = 1: log(5m) * log(m) = log(2)^log(10) then, you have log(5m) * log(m) = log(10) * log(2) log(5m) * log(m) = log(5 * 2) * log(2) then m = 2, 'cause the expressions on both sides are the same so: x - 5 = 2 x = 7

@superacademy247 6 дней назад

Yes. And there's one more solution x=5.1

@tristan583 7 дней назад

Great but results need verification, X can’t have two different acceptable values

@masoudghiaci5483 17 дней назад

❤❤

@miffyLake 21 день назад

That`s great teacher.

@KT-vi7gd 14 дней назад

Detailed explanation ! Good !

@superacademy247 14 дней назад

Glad it was helpful!

@naakatube 3 дня назад

Solved it in my head in 30 sec!

@superacademy247 2 дня назад

Can you solve it overhead for the second solution?

@umutkargili3617 22 дня назад

Appriciated, that's why computer is invented

@RealQuInnMallory 14 дней назад

(5x)^2(25x)= {25x^2 625 x^2}= 600x (x (minus ➖ 3x+2)

@joserubenalcarazmorinigo9540 17 дней назад

Excelente, pero en el discriminante también se puede cambiar 2 = 10/5 y trabajar con log 5, entonces: log2 = log(10/5)=log10-log5=1-log5

@yifengxiao 14 дней назад

multiplying both sides by 5, then let y=5x-25. we have (logy)^2=1, ....

@SidneiMV 15 дней назад

ln(x - 5)[ln(5x - 25)]/ln10 = ln2 ln(x - 5) = u u(u + ln5) = (ln2)(ln2 + ln5) u² + uln5 - (ln2)(ln2 + ln5) = 0 u = (-ln5 ± √[ln²5 + 4ln²2 + 4(ln2)ln5]/2 u = [-ln5 ± (ln5 + 2ln2)]/2 u = ln2 => ln(x - 5) = ln2 => *x = 7* u = -ln10 => x - 5 = 1/10 => *x = 5.1*

@user-ee7nw2rx9s 21 день назад

А проще нельзя? Логарифмировать по основанию 10 Получаем log(x-5)* log(5*(x-5))=log2, Есть формула log (a*b) =loga+log b Отсюда log (x-5)*(log(x-5)+log5) =log2 И замена log (x-5)=a Получаем обычое квадратное уравнение а*(а+log 5)=log2

@victorfildshtein 14 дней назад

Тут фишка не в том, чтобы привести к квадратному уравнению, а в том, чтобы извлечь корень из дискриминанта.

@user-ee7nw2rx9s 10 дней назад

Дискриминант (log5)^2-4 log2 Если учесть что 10=5*2, а log 10=1 и log(a*b) =loga+logb, то Получаем полный квадрат, как в решении и гораздо быстрее

@ludmilak9396 21 день назад

Надо было сразу указать, что log -- это ln, то есть логарифм по основанию 10.

@mmdexperiments3828 21 день назад

ln - это натуральный логарифм, по основанию е. А логарифм по основанию 10 пишется lg. Согласна, что тут непонятно в условии, что за логарифм, т.к. написано просто log без основания. Так что либо автор ошибся, либо это местная форма записи такая, мол, без основания это значит по 10. Но я сомневаюсь.

@user-bs8mi3qw7g 21 день назад

This is to challenge our brain and math concepts. Dependence on computers is for people who don’t think for themselves. It’s for zombies! Dr. K

@kevinmadden1645 20 дней назад

AKA Democrats.

@kevinmadden1645 20 дней назад

You could read "War and Peace" three times cover-to-cover and he would not have solved this problem .After logging both sides to base ten, work with ratios. Much easier!