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

Expert Answers info

justaguide eNotes educator | Certified Educator

calendarEducator since 2010

write12,544 answers

starTop subjects are Math, Science, and Business

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

check Approved by eNotes Editorial

giorgiana1976 | Student

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

check Approved by eNotes Editorial

Unlock This Answer Now