Think about the function \(y=10^x\). Every time we increase \(x\) by one, we multiply \(y\) by \(10\).

By how much do we multiply \(y\) when we increase \(x\) by \(0.5\)?

Given that \(\sqrt{10}\approx 3.16\) and \(\sqrt[4]{10}\approx 1.78\), complete the table below.

\(10^0=\) \(10^{0.25}\approx\) \(10^{0.5}\approx\) \(10^{0.75}\approx\)
\(10^1=\) \(10^{1.25}\approx\) \(10^{1.5}\approx\) \(10^{1.75}\approx\)
\(10^2=\) \(10^{2.25}\approx\) \(10^{2.5}\approx\) \(10^{2.75}\approx\)
\(10^3=\) \(10^{3.25}\approx\) \(10^{3.5}\approx\) \(10^{3.75}\approx\)

What does the whole number part of the power of ten tell us about the value of \(y\)?

How is this connected with standard form?

Can you now find approximate solutions to the following equations?

  1. \(10^x = 21\)

  2. \(10^x = 5\,000\)

  3. \(10^x = 562\,000\)

  4. \(10^x = 0.00178\)

See the resource 1950s calculators for a further exploration of this.