x=Log(10)/Log(-10). Log is the complex logarithm. The principle values of Log(10) and Log(-10) are log(10) and log(10)+iπ respectively, where log is the natural logarithm.
One is only allowed to use the representations -1 = exp(πi) when taking the logarithm; it is always the case that 1^x = 1, whatever the value of x. Thus the equation becomes exp((ln(10) + πi)x) =10. At this point we are allowed to use the fact that exp(2πni) = 1 to find all possible solutions x (when n is an integer). Hence we can write the right hand side as exp(ln(10) + 2πni), and x = (ln(10) + 2πni)/(ln(10) + πi), n = 0, ±1, ±2, ....
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x=Log(10)/Log(-10).
Log is the complex logarithm. The principle values of Log(10) and Log(-10) are log(10) and log(10)+iπ respectively, where log is the natural logarithm.
One is only allowed to use the representations -1 = exp(πi) when taking the logarithm; it is always the case that 1^x = 1, whatever the value of x. Thus the equation becomes
exp((ln(10) + πi)x) =10.
At this point we are allowed to use the fact that exp(2πni) = 1 to find all possible solutions x (when n is an integer). Hence we can write the right hand side as exp(ln(10) + 2πni), and
x = (ln(10) + 2πni)/(ln(10) + πi), n = 0, ±1, ±2, ....
x=ln10/(ln10+iπ)