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 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
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Since the bases are primes and they are not matching, we can use logarithms to solve exponential equations.
We'll take logarthims both sides:
log13 [13^(13x+1)] = log13 (3^x)
We'll apply the power rule for logarithms:
(13x+1) log13 13 = x log13 3
We'll recall that log13 13 = 1
We'll re-write the equation:
13x+1 = x log13 3
We'll subtract x log13 3 both sides to isolate x terms to the left side.:
13x - x log13 3 = -1
We'll factorize by x:
x(13 - log13 3) = -1
We'll re-write log13 3 = lg3/lg13
x(13 - lg3/lg13) = -1
We'll divide by 13 - lg3/lg13 = 12.5702
x = -1/12.5702
Rounded to four decimal places:
x = -0.0795
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