12^(x+10)=13^(-9x) Write the exact answer using either base-10 or base-e logarithms.

Answer 1

(x+10)log12=-9xlog13 Just move the exponent into a coefficient of the log

(x+10)/(x)=-9log13/log12 divide by xlog 12

1+10/x =- 9log13/log13 reduce left side to 2 terms

10/x =- 9log13/log12- 1 subtract 1 from both sides

x = 10/(-9log13/log12 -1) is the exact answer which is ever so slightly larger than -1

log12 is slightly smaller than log 13 so x = 10/(-9(1.03223)-1) = -0.972...for 3 decimal places

This is using base 10

check the answer, plug in x=-1 into the original equation

to get 12⁹ = 13⁹ You need ever so slightly bigger than -1. -0.99 is closer, but -0.972..is closest