Question

If 4^a«8^b What is the ratio of a to b?

Answer 1

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 \( \frac{3}{2} \).