Can one use logarithms to solve the equations 2=3^x+x and 2=3^x x? Could someone explain how would you solve: 2=3^x+x and 2=3^x⋅x I can only solve halfway through. And why is 10^(log(x))=x Thanks

HypeMyday3m 2022-09-25 Answered
Can one use logarithms to solve the equations 2 = 3 x + x and 2 = 3 x x?
I can only solve halfway through.
And why is
10 log ( x ) = x
Thanks
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Answers (1)

Genesis Rosario
Answered 2022-09-26 Author has 11 answers
By definition, a log is a quantity representing the power to which a fixed number (the base) must be raised to produce a given number.
Below is a simple example,
10 2 = 100
So
log 10 100 = log 10 10 2 = 2 log 10 10 = 2
And
10 log 10 100 = 10 2 = 100
Generally
b log b ( x ) = x
Also, the solutions to both of those problems cannot be found using elementary functions.
To solve these equations, we must use the Lambert W function. This function will provide the value of x in equations that take the form z = x e x
z = x e x W ( z ) = x
Since you're curious, here are the solutions
2 = 3 x x = x 3 x = x e ln 3 x = x e x ln 3
2 ln 3 = ( x ln 3 ) e x ln 3
W ( 2 ln 3 ) = x ln 3
x = W ( 2 ln 3 ) ln 3
And after some trial and error,
2 = 3 x + x
2 x = 3 x
( 2 x ) 3 2 = 3 x 3 2
( 2 x ) 3 2 3 x = 9
9 = ( 2 x ) 3 2 3 x = ( 2 x ) 3 2 x = ( 2 x ) e ln 3 2 x = ( 2 x ) e ( 2 x ) ln 3
9 ln 3 = ( ( 2 x ) ln 3 ) e ( 2 x ) ln 3
W ( 9 ln 3 ) = ( 2 x ) ln 3 = 2 ln 3 x ln 3
W ( 9 ln 3 ) 2 ln 3 = x ln 3
x = 2 ln 3 W ( 9 ln 3 ) ln 3
Did you like this example?
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