Solve. Check for inadmissible roots.

Question

5^x = 7-12(5^-x)

Answer 1

Let's multiply everything by 5^x. Remember that when two quantities have the same base, you can add their exponents (a^b × a^c = a^b+c).

 

5^x(5^x) = 5^x(7 - 12(5^-x))

(5^x)² = 7(5^x) - 12(5^x-x)

(5^x)² = 7(5^x) - 12(5⁰)

(5^x)² = 7(5^x) - 12

 

 

 

Let 5^x = y. Then we've got a quadratic equation:

 

y² = 7y - 12

y² - 7y + 12 = 0

(y - 3)(y - 4) = 0

y - 3 = 0    OR    y - 4 = 0

y = 3    OR    y = 4

 

 

 

Now, substitute 5^x back in for y:

 

5^x = 3    OR    5^x = 4

 

 

 

Finally, change those into log form to solve:

 

log₅3 = x

0.68 = x

 

log₅4 = x

0.86 = x

 

 

 

Plug those values of x back in to make sure they work.