Question

a. Create a sort of complex exponential equation using a common base and solve.b. Then solve the exponential equation using logarithms.

Answer 1

We can solve a common base exponential equation of the form:

$a^{f(x)}=b^{g(x)}$

With the one-to-one property to get:

$f(x)=g(x)$

(a) In this case, we can take 4 as the common base, to get:

$4^{f(x)}=4^{g(x)}$

For f(x), we can use a linear equation of the form mx + b like 6x + 9, and for g(x) we can make it equal to -x² (we can use whatever equation we want because we are creating the complex exponential equation), then we get:

$4^{6x+9}=4^{-x^2}$

Since we have common bases, by means of the one-to-one property, we rewrite the above equation to get:

$6x+9=-x^2$

Simplifying and factoring:

6x + 9 + x² = x² - x²

6x + 9 + x² = 0

(x + 3)² = 0

Then, the solution to this equation is -3

(b) Similarly, taking logarithms on both side we get:

$\begin{gathered} Log(4^{6x+9})=Log(4^{-x^2}) \\ (6x+9)Log(4^{})=(-x^2)Log(4^{}) \end{gathered}$

By dividing both sides by Log(4), we get:

$6x+9=-x^2$

As you can see, we got the same equation as in part (a), then the solution will be the same and it is x = -3