Write the exact answer using either base-10 or base- e logarithms.

Question

17^-x+10 = 14^7x

Write the exact answer using either base-10 or base- e logarithms.

Answer 1

17^-x+10 = 14^7x is a little ambiguous

it could mean

17^(-x+10) = 14^7x

or as written it's

(17^-x)+10 =14^7x

the 1st interpretation leads to

17^(-x+10)=14^7x

take logs of both sides

(-x+10)ln17 =7xln14

-xln17 - 7xln14 =-10ln17

x = -10ln17/[-ln17-7ln14]

x= exactly 10ln17/(ln17+7ln14

=about 28.33/(2.83+18.47)

x= about 1.32973...

try an equation solver calculator

check the answer

17^(-1.33+10) = 14^(7(1.33)

17^8.67 = 14^9.31

4.66 x 10^10 = 4.68 x 10^10

close enough, given rounding approximations

the 2nd interpretation leads to another answer

try graphing the equation(s) and see where the curve intersects the x axis seems

about x= .7 or .1 for the two equations, which

check .7

17^(-.7+10) = 14^(.7(7)

17^(9.3) = 14^4.9

2.8 x 10^11 doesn't come close to 4.1 x 10^5

maybe

try a different graphing calculator