First isolate the power of 10 on the right side by dividing both sides by 5:
[tex]100=5\cdot10^{0.3t}\implies20=10^{0.3t}[/tex]
Now you're free to take the logarithm and simplify where you can:
[tex]\log_{10}20=\log_{10}10^{0.3t}=0.3t\log_{10}10=0.3t[/tex]
Then solve for [tex]t[/tex]:
[tex]t=\dfrac{\log_{10}20}{0.3}\approx4.34[/tex]
