Not sure how to solve this problem without graphing

[IntegralDerivative]

Homework Statement

Find the solution(s) when: 4+x = 4(2)^x

Homework Equations

The Attempt at a Solution

I only know how to solve this problem using graphing. I'm not sure how to do it algebraically. Please help.

 

[Fred Wright]

I suggest that you use the fact that,$$2^x=e^{xln\left (2\right )}$$
and expand the exponential.

 

[scottdave]

Some problems do not have a nice easy algebraic solution. The (0,4) is easy enough to guess and also verify.

 

[Ray Vickson]

Homework Statement

Find the solution(s) when: 4+x = 4(2)^x

Homework Equations

The Attempt at a Solution

I only know how to solve this problem using graphing. I'm not sure how to do it algebraically. Please help.

View attachment 203610

As scottdave has indicated, some problems do no have nice solutions, and this is one of them. However, some have not-so-nice solutions, as does this one:
$$x = 0 \; \text{or} \; x = -4 - \frac{1}{\ln 2} \text{LambertW} \left( -\frac{1}{4} \ln 2 \right) \doteq -3.690093068 $$
Here, LambertW is a non-elementary function that does not have an explicit, finite formula, but has known series expansions and can be approximated numerically to as much accuracy as you want. You will not find it in a spreadsheet, but it is available in computer algebra systems such as Maple and Mathematica.

 

[Buffu]

As scottdave has indicated, some problems do no have nice solutions, and this is one of them. However, some have not-so-nice solutions, as does this one:
$$x = 0 \; \text{or} \; x = -4 - \frac{1}{\ln 2} \text{LambertW} \left( -\frac{1}{4} \ln 2 \right) \doteq -3.690093068 $$
Here, LambertW is a non-elementary function that does not have an explicit, finite formula, but has known series expansions and can be approximated numerically to as much accuracy as you want. You will not find it in a spreadsheet, but it is available in computer algebra systems such as Maple and Mathematica.

Can "Lambert W function" give $x= 0$ as the solution ? without guessing ?

 

[ehild]

Homework Statement

Find the solution(s) when: 4+x = 4(2)^x

Homework Equations

The Attempt at a Solution

I only know how to solve this problem using graphing. I'm not sure how to do it algebraically. Please help.

View attachment 203610

You can apply some numerical methods. One is an iterative method (http://tutorial.math.lamar.edu/Classes/CalcI/NewtonsMethod.aspx)
You should write the equation to be solved in the form x=f(x). It works in the domain where |f '(x)|<1
The problem becomes very simple if you switch to the variable y = 2+x. The equation becomes y=2^y-2. It converges quite fast for y<0.
Substitute some initial value for y, you get the next value, y₁. Substitute y₁, you get the next approximation, y₂.
Continue till |y_n+1-y_n| are close enough.
Starting from y= 0, we get -1, -1.5, -1.646, -1.680, -1.688, -1.689, -1.690...which corresponds to x=-3.690.
The derivative of 2^y-2= 2^ylog(2) is greater then 1 for y>0.529. To get the other root (y=2), an other iterative function is needed: y=log(y+2)/log(2).

 

[Ray Vickson]

Can "Lambert W function" give $x= 0$ as the solution ? without guessing ?

Yes. The other solution given by Maple is
$$x = -4 = \frac{1}{\ln 2} \text{LambertW}\left( -1, -\frac{1}{4} \ln 2 \right), $$
where $\text{LambertW}(-1,z)$ is one of the non-analytical branches of the Lambert function, chosen so that it is real-valued on the real interval $(-e^{-1}+i 0, 0 + i 0)$ in the complex $z$-plane. It is not easy to see, but Maple evaluates this as $x = 0$ exactly. Of course, a person would see $x=0$ right away, without use of any advanced tools.

 

[IntegralDerivative]

Wow thank you so much guys. :)

 

[scottdave]

The article in lamar.edu posted by @IntegralDerivative is a nice one, explaining numerical methods.

 

[SammyS]

The article in lamar.edu posted by @IntegralDerivative is a nice one, explaining numerical methods.

Actually that was posted by my friend, @ehild .

 

[ehild]

Actually that was posted by my friend, @ehild .

Thank you friend