Express $(100^3)^5$ with a base of $10$

Question

Express $(100^3)^5$ with a base of $10$. I don't get this.

Answer 1

First let's combine the exponents by using the rule $(x^a)^b = x^{a\cdot b}.$ Thus we have $$(100^3)^5 = 100^{3 \cdot 5} = 100^{15}.$$ Next, we note that $100 = 10^2$. So we replace $100$ in the above equation with $10^2$ and apply the same rule.$$100^{15} = (10^2)^{15} = 10^{2\cdot 15} = 10^{30}.$$ Thus, we've expressed it with base 10. Let me know if you have any other questions!

Answer 2

It means going for this representation $$ (100^3)^5 = 10^x \quad (*) $$ The left hand side is $$ (100^3)^5 =100^{15} = (10^2)^{15} = 10^{30} $$ Or we take the logarithm of both sides of $(*)$: $$ \ln((100^3)^5) = \ln(10^x) \iff \\ 5 \ln(100^3) = 15 \ln(100) = x \ln(10) \iff \\ x = 15 \ln(100)/\ln(10) = 30 \ln(10)/\ln(10) = 30 $$