Solve the expression 13^(13x+1)=3^x for x

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We have to solve 13^(13x+1)=3^x for x.

Looking at 13^(13x+1)=3^x, the bases are not the same, so we cannot equate the exponential.

Therefore, we can only solve the problem using logarithms.

Take the log of both the sides of the equation

log 13^(13x+1)= log 3^x

=> (13x + 1)* log...

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We have to solve 13^(13x+1)=3^x for x.

Looking at 13^(13x+1)=3^x, the bases are not the same, so we cannot equate the exponential.

Therefore, we can only solve the problem using logarithms.

Take the log of both the sides of the equation

log 13^(13x+1)= log 3^x

=> (13x + 1)* log 13 = x* log 3

=> 13x * log 13 + log 13 = x log 3

=> 13x * log 13 - x log 3 = - log 13

=> x( 13 log 13 - log 3) = -log 13

=> x = -log 13 / ( 13 log 13 - log 3)

=> -0.0795 approximately

So x = -log 13 / ( 13 log 13 - log 3) or -0.0795

Approved by eNotes Editorial Team