Question

solve the exponential equation. express the solution set so that (a) solutions are in exact form and, if irrational, (b) solutions are approximated to the nearest thousandth. support your solutions by using a calculator.

Answer 1

You have the following equation:

`((1)/(8))^x=-3`

In order to solve the previous equation for x, proceed as follow:

- write 1/8 as 8⁻¹ and simplify exponents:

`((1)/(8))^x=(8^(-1))^x=8^(-x)`

Then, the equation becomes:

`8^(-x)=-3`

next, use properties of logarithms to obtain x. In this case apply log with base 8 to cancel 8, as follow:

`\log _8(8^(-x))=-\log _88^x=-x`

and the equation becomes:

`-x=\log _8(-3)`

Then, you obtain log_8 (-3). Due to logarithms of negative numbers do not exist, then, the equation does not have solution for x.

Hence, in both (a) and (b) cases you have:

The solution is the empty set