Question

What is the approximate solution of 3^x-1 = 8.2?

Answer 1

Any choices?

Answer 2

is it \[\huge 3^{x-1}=8.2\]?

Answer 3

x=3.06

Answer 4

unlikely

Answer 5

Yes @satellite73

Answer 6

\(3^{1} = 3\)
\(3^{2} = 9\)

The question is no good. "The approximation" doesn't exist. "AN" approximation can be generated.

Linear Interpolation will produce AN approximation: \(1 + \dfrac{8.2-3}{9-3} = 1.8667\), Thus "AN" approximation for x is 2.8667.

There are infinitely many approximations.

One might also observe, \(3^{x-1} = \dfrac{3^{x}}{3} = 8.2 \implies 3^{x} = 24.6\)

\(3^{3} = 27\)

And we have another Linear Interpolation will produce AN approximation: \(2 + \dfrac{24.6-9}{27-9} = 2.8667\), Thus "AN" approximation for x is 2.8667. It should be no surprise that these produce the same result.

Any other ideas? Calculus? Iteration of some sort?

Answer 7

Refer to the attachment.