If \(a^2 = \dfrac{b^3}{10}\) and \(\log_{10} a = \log_{10} b+1\), find the values of \(a\) and \(b\), expressing each answer as a power of 10.

Approach 1:

We have that \(10a^2 = b^3\). Taking logs to base \(10\) gives \(\log_{10}10 + 2\log_{10}a=3\log_{10}b.\)

But we know that \(\log_{10} a = \log_{10}b + 1\), and so \(1+2(\log_{10}b + 1)=3\log_{10}b.\)

This gives \(\log_{10}b = 3\), and so \(b = 10^3\), and \(a = 10^4\).

Approach 2:

We have \[\begin{align*} \log_{10} a &= \log_{10} b+1\\ \implies \log_{10} a &= \log_{10} b + \log_{10} 10\\ \implies \log_{10} a &= \log_{10} (10b)\\ \implies a &= 10b. \end{align*}\]

Using the other equation, we find \[100b^2 = \frac{b^3}{10},\] and so \(b=0\) or \(b=1000\).

However \(\log_{10} 0\) is not defined, and hence \(b=0\) is not a solution.

Thus \(b=10^3\), which leads to \(a=\pm 10^4\). Again, \(\log_{10}a\) appears in the question so \(a\) cannot be negative.

So we have \(b=10^3\) and \(a=10^4\).