Is there an easy way to find parameter by three information?

Question

$f(x)=a+b.c^x$ and we know $$f(0)=15\\ f(2)=30\\f(4)=90$$ so put down point in function and have: $$a+b.1=15\\a+b.c^2=30\\a+b.c^4=90$$so $$(2)-(1) \to bc^2-b=30-15\\(3)-(2) \to bc^4-bc^2=90-30\\b(c^2-1)=15\\bc^2(c^2-1)=60 \to \div\\ \frac{bc^2(c^2-1)}{b(c^2-1)}=\frac{60}{15}\\c^2=4\\c=\pm 2 \\\ \to c=+2\to a=10,b=5$$ Is there was a way to find $a,b,c$ easier or by other method ,for example using logarithm ...?