Exponential Equation With Double Exponents | How To Solve Exponential Equations | Math Olympiad

In this math algebra video, we shall solve an exponential equation with double exponents.

[cmilkau]

Number theorists sees this, writes down "no integer solutions" and moves on.

[TherealtalktimesphrmdrMagumbas]

You are a real master of math.
Thanks sir

[walterwen]

Math Olympiad, Solve Double Exponential Equation: 3^(5^x) = 5^(3^x)
Firstly, convert the exponential base 5 into 3 using the logarithmic math method:
Let 3^n = 5, n = log5/log3 = 0.699/0.477 = 1.465; 5 = 3^1.465
3^(5^x) = 5^(3^x) = (3^1.465)^(3^x) = 3^[(1.465)(3^x)]
5^x = (1.465)(3^x), log(5^x) = log[(1.465)(3^x)]
xlog5 = log1.465 + xlog3, x = log1.465/(log5 – log3) = 0.166/0.222 = 0.747
Answer check:
3^(5^x) = 3^(5^0.747) = 3^3.328 = 38.7
5^(3^x) = 5^(3^0.747) = 5^2.272 = 38.7; Confirmed

[walterwen]

Math Olympiad, Solve Double Exponential Equation: 3^(5^x) = 5^(3^x)
Another method:
Convert the exponential base 3 and 5 into 10 using the logarithmic math method:
log3 = 0.477; 3 = 10^0.477, log5 = 0.699; 5 = 10^0.699
3^(5^x) = (10^0.477)^(5^x) = 10^[(0.477)(5^x)]
5^(3^x) = (10^0.699)^(3^x) = 10^[(0.699)(3^x)]
(0.477)(5^x) = (0.699)(3^x)
(0.477)(5^x) = (0.477)(10^0.699x) = (0.699)(3^x) = (0.699)(10^0.477x)
(10^0.699x)/(10^0.477x) = 0.699/0.477 = 1.465
(10^0.699x)/(10^0.477x) = 10^(0.699 – 0.477x) = 10^0.222x = 1.465
0.222x = log1.465 = 0.166, x = 0.166/0.222 = 0.747
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