Three minor comments: 5 ± 2√6 is a perfect square so log(5 ± 2√6) can be further "simplified" to 2*log(√3 ± √2). The change of base is often done but I prefer leave it as base 10 or e. In this case the result would be 2*log(√3 ± √2)/(log 2 - log 5). Suit yourself. I observe that 5 - 2√6 is the inverse of 5 + 2√6, i.e. they're conjugates whose product is 1. So log(5 - 2√6) is the negative of log(5 + 2√6) and the two final results are the same except for the sign. I enjoy your interesting problems.
Three minor comments:
5 ± 2√6 is a perfect square so log(5 ± 2√6) can be further "simplified" to 2*log(√3 ± √2).
The change of base is often done but I prefer leave it as base 10 or e. In this case the result would be 2*log(√3 ± √2)/(log 2 - log 5). Suit yourself.
I observe that 5 - 2√6 is the inverse of 5 + 2√6, i.e. they're conjugates whose product is 1. So log(5 - 2√6) is the negative of log(5 + 2√6) and the two final results are the same except for the sign.
I enjoy your interesting problems.
Thanks 😊 for your input 👍💯 and support. Great analysis. Very brilliant.
If Both side were divided by 25^x the result will be faster, for my opinion. Explanation was very good. All explanations comments were very nice.
Thank you 💕 very much
❤❤❤❤
Give the reference to the olympiad it is from. This is too easy to be from an olympiad
China Math Olympiad Problem: 4ˣ + 25ˣ = 10ˣ⁺¹; x = ?
4ˣ > 0; (4ˣ + 25ˣ)/4ˣ = (10ˣ⁺¹)/4ˣ, 1 + (25/4)ˣ = 10(10/4)ˣ
1 + (5/2)²ˣ = 10(5/2)ˣ; [(5/2)ˣ]² - 10(5/2)ˣ = - 1, [(5/2)ˣ - 5]² = 25 - 1 = 24
[(5/2)ˣ - 5]² = (2√6)², (5/2)ˣ = 5 ± 2√6 > 0, x = log(5 ± 2√6)/log(5/2)
5 + 2√6 = 5 + 4.899 = 9.899; x = log9.899/log2.5 = 0.996/0.398 = 2.503
5 - 2√6 = 5 - 4.899 = 0.101; x = log0.101/log2.5 = - 0.996/0.398 = - 2.503
Answer check:
x = 2.503: 4ˣ + 25ˣ = 4²·⁵⁰³ + 25²·⁵⁰³ = 32.133 + 3155.323 = 3187.5
10ˣ⁺¹ = 10³·⁵⁰³ = 3184.2; Confirmed
x = - 2.503: 4⁻²·⁵⁰³ + 25⁻²·⁵⁰³ = 0.0311 + 0.0003 = 0.0314
10ˣ⁺¹ = 10⁻¹·⁵⁰³ = 0.0314; Confirmed
The calculation was achieved on a smartphone with a standard calculator app
Final answer:
x = 2.503 or x = - 2.503
I.got it but explanation is too big. Any short cut?
There are various methods. I'll do a video for the shortcut method
x=ln(5±2vʼ6)/(ln2-ln5)≈±(-2.502) 😁