Solve for x in 2^x-5.2^(x+1)=-144 when the x is an exponents applied to different bases?

Question

Question: Find X in 2^x -5.2^x+1 = -144

My Attempt to the solution:

2^x -5.2^x+1 = -144

2^x-5.2^x-5.2¹ = -144

2^x-5.2^x = -144 + 5.2

How would you solve for x when two bases are different?

Answer:

2^x(1-5.2) = -144

2^x(-9) = -144

2^x = 16

x = 4

Also, please tell me what I did wrong in my solution and how where the -9 in the second line under answer came from.

Answer 1

You mistake is treating $2^{x+1} = 2^x + 2$ which is not true.

It should be $2^{x+1}=2\cdot 2^x$.

$-9$ comes from $1-2\cdot 5 = 1-10=-9$

Edit:

Further mistake:

Even if you interpret the question as $2^x +(-5.2)^{x+1}=-144$, $$2^x +(-5.2)^{x}(-5.2)=-144$$

This is not equivalent to $$2^x-5.2^x-5.2=-144$$

We can't just convert multiplication to addition. '

Remark: If the quesiton is really $$2^x-5.2^x = -144$$

Notice that $2^x-5.2^x$ is a decreasing function for $x>0$, you might want to use bisection method to find the numerical solution.

Answer 2

$2^{x+1}=2*2^x $.

So the rest should come naturally.

$2^x-5*2^{x+1}=-144$

$2^x-5*2*2^x=-144$

$2^x-10*2^x=-144$

$2^x (1-10)=-144$

$2^x*(-9)=-144$

$2^x=\frac {-144}{-9}=16$

$2^x=2^4$

$x=4$

And your mistake was thinking $5*2^{x+1}=5*2^x+5*2^1$. Powers don't distribute over addition. They distribute over multiplication. $2^{x+1}\ne 2^x+2^1$. But $2^{x+1}=2^x*2^1$ instead.