To compare \(4^a\) and \(8^b\), we can express both numbers in terms of powers of \(2\).
First, we know that:
\[ 4 = 2^2 \quad \text{and} \quad 8 = 2^3 \]
Thus, we can rewrite \(4^a\) and \(8^b\) as follows:
\[ 4^a = (2^2)^a = 2^{2a} \] \[ 8^b = (2^3)^b = 2^{3b} \]
Now we can set up the inequality \(4^a < 8^b\) in terms of powers of \(2\):
\[ 2^{2a} < 2^{3b} \]
Since the bases are the same (base \(2\)), we can compare the exponents:
\[ 2a < 3b \]
Now, to find the ratio of \(a\) to \(b\), we can rearrange the inequality:
\[ \frac{2a}{3b} < 1 \]
This implies:
\[ \frac{a}{b} < \frac{3}{2} \]
So, the ratio of \(a\) to \(b\) is:
\[ \frac{a}{b} < \frac{3}{2} \]
Thus, the ratio of \(a\) to \(b\) is less than \( \frac{3}{2} \).
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