3^(x-2) = 5^x?

Question

i tried solving this with logarithms but i think i dont know enough of the properties of logarithms to do this properly

find the exact solution

Answer 1

xln(3) - 2ln(3) = xln(5)
x(ln(5) - ln(3)) = - 2ln(3)
x = - 2ln(3) / ln(5/3)
x = - 4.301320

check:
3^-6.301320 =? 5^-4.301320
9.8515869968783440018802465316641e-4 = 9.8515869968783440018802465316641e-4
to 32 significant digits.

Exact solution in simplest terms:
x = (ln(9)) / ln(0.6)

Rounded to 32 significant digits:
x ≈ -4.30132 02061 74247 01770 95956 84581 4

Answer 2

Answers with Solutions

Qns. 1:

3^x / 3^2 = 5^x
lg 3^x / lg 3^2 = lg 5^x
lg 3^x / lg 5^x = lg 3^2
x lg 3 - x lg 5 = lg 9
x (lg 3 - lg 5) = lg 9
x = lg 9 / (lg 3 - lg 5)
= - 4.301
≈ - 4.3

Hope This Solutions Will Help In Answering Your Questions ;)

G0n9 G0n9™
Mathematician

Answer 3

3^(x-2)=5^x
(x-2)ln3=xln5

hence x=2ln3/(ln3-ln5)

'ln' here means log to any base ,not necessarily e

Answer 4

(x-2)ln3=xln5
xln3-2ln3=xln5
xln3-xln5=2ln3
x(ln3-ln5)=2ln3
x=(2ln3)/(ln3-ln5)

im assuming you were not distributing the ln3 thru, thats the only messup i could see.

Answer 5

(x-2)ln3=xln5
xln3-2ln3=xln5
xln3-xln5=2ln3
x(ln3-ln5)=2ln3
x=(2ln3)/(ln3-ln5)
x=(ln9)/ln(3/5)

x=ln(9-(3/5))

x=ln(43/5)

If u want accurate answer 43/5=8.6
ln8.6=2.1517622

x=2.1517622

Answer 6

3^(x-2) = 5^x
Take log on both sides
log 3^(x-2) = log 5^x
(x-2)log3=xlog5..since log a^b=bloga
xlog3-xlog5=2log3
x(log3-log5)=log3^2
xlog(3/5)=log9.................log(a/b)=loga-logb
xlog(0.6)=log9
x=log9/log0.6
x=0.954/(-0.222)
x= - 0.4297

Answer 7

x = -4.301
try it

Answer 8

???