Question

How do you solve `3^x=243`?

Answer 1

`5`

Explanation:

Take logs of both sides, to any convenient base.

`ln(3^x)=ln(243)`
Hence
`x ln(3)=ln(243)`
`x=ln(243)/ln(3)'`
`approx5`

This used logs to base `e` (ln(...), natural logarithms). You get the same final answer if you use logs to base 10 (log(...), common logarithms ).

If your calculator does logs to any base you could have used base 3 which would have immediately given you `log_3(243)=5`

Check by direct multiplication `3^5=243`, so the approximation above is in fact exact, as you would suspect having found that the display on the calculator was either exactly `5` or extremely close.