Building blocks

## Problem

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.