HOW do you solve this type of equation? (e^x)^x * e^54 = e^15x

Question

(e^x)^x _* e⁵⁴ = e^15x

The answer: (9,6)

I missed this lecture in class because I had no other choice. Please help.

Answer 1

The majority of the time when dealing with e power , we use natural logarithm to solve this type of equations.

1.e^x)^x *. e^54. = e^15x

2. Dividing both sides by. e^54. , I got (e^x)^x. = (e^15x) /(e^54)

3. Subtracting the exponents of e^15x and e^54. in the right side side , I got e^15x-54

4 . (e^x)^x = e^15x-54

5 . Multiplying the exponents in the left side. (e^x)^x= e^x2

6 e^(x^2). = e^15x-54 Applying natural log in both sides.

7. ln e^x^2. = ln e^15x-54 power property of log. ln A^x = x ln A

8 X^2 ln e. =15x-54 ln e lne =1. you can verify this in your calculator .

9. x^2 = 15x-54 grouping the terms in the left side

10 x^2-15x + 54 =0. Factoring. (two numbers -9 x -6 =+54 and -9 + -6 =. -15

11. (x - 9 ) ( x-6 ) =0. Solving for x using property of zero.

12. x-9 =0. , x=9

13 x-6 =0. , x=6

The solution is (9 and 6).

To check substitute thee values in the original equation (e^9)^9 * e^54=e^15(9)

e^81. *. e^54 = e^135

e^135 =e^135

For x=6 (e^6)^6. * e^54 = e^ 15(6)

e^36. * e^54. = e^90

e^90 =e^90 Both answers check the original equation.

Answer 2

Hi Jac,

We start with the property that when we raise an exponent to another exponent, we multiply the two together:

(e^x)^x = e^x*x = e^{x^2}

Next the property that when we multiply two numbers together with the same base, we can add the exponents:

e^{x^2} * e⁵⁴ = e^{x^2 + 54}

Now since the left and right hand sides both have e as the base, we can ignore those e’s and set the exponents equal to each other:

x² + 54 = 15x which means

x² - 15x + 54 = 0

(x - 9)(x - 6) = 0

x = 9, 6

Please let me know if I can be more helpful, thanks!

Joe