10,000= 20,000 (1.3) ^ -0.234t?

You have to use logs

First divide by 20000 to get 1/2 = (1.3)^-.234t
then log both sides (either log or ln it doesn't matter)
log (1/2) = log(1.3)^-.234t
according to the log rules, the log of a power is the power times the log (the power comes to the front)
log (1/2) = -.234t log(1.3)
divide both sides by log 1.3 to get
-.26419... =-.234t
divide by -.234
11.29 = t (rounded to nearest hundredth, round where you need to)

1/2 = 1.3^(- 0.234 t )

ln 0.5 = (- 0.234 t ) ln 1.3

- [ ln 0.5 / ln 1.3 ] = (0.234 t )

t = 11.3

10000 = 20000(1.3)^-0.234t
1/2 = 1.3^-0.234t
log_1.3(1/2) = log_1.3(1.3^-0.234t)
-0.234t[log_1.3(1.3)] = log_1.3(1/2)
-0.234t = -2.642
t = -2.642/(-0.234)
t = 11.29

10000 = 20000 (1.3)^-0.234t
0.5 = 1.3^-0.234t
lg 0.5 = -0.234t lg 1.3
lg 0.5/lg 1.3 = -0.234t
t = 11.3 (3 significant figures)

10,000= 20,000 (1.3) ^ -0.234t

Divide both sides by 20,000
0.5 = 1.3 ^ -0.234t

Take the log of both sides (natural log, log base e, ln)
ln(0.5) = ln(1.3 ^ -0.234t)

Using log laws,
ln(0.5) = (-0.234t)ln(1.3)

Divide both sides by ln(1.3)
ln(0.5)/ln(1.3) = -0.234t

Divide both sides by -0.234
[ln(0.5)/ln(1.3)]/-0.234 = t
t = 11.290