solve function for log?

I am trying to solve the following equation for y, where x and k are known

log(y)+log(((k-1-x(i))/(k-1-y*x(i)))^k)== log(4)

when I use solve function, it returns a warning of spurious solution, and subscripted assignment mismatch error.

the program works if I get rid of the k in the power

i.e

 log(y)+log(((k-1-x(i))/(k-1-y*x(i))))== log(4)

I tried using vpasolve function too, but it did not help.

Any help will be appreciated

Here is the code:

clc; clear all;
syms y;
k=50;

x = 0.01:1:10;
[~,n] = size(x);

for i= 1:n;
    eqn=log(y)+log(((k-1-x(i))/(k-1-y*x(i)))^k)== log(4);
    z(i)=10*log10(solve(eqn,y));    
end

figure(1)
set(gca, 'XScale', 'log')
semilogx(x,z)

Answer 1

[Star Strider]

You need to calculate a numeric solution to your equation. One way to understand what it is doing is to find and plot the contour of the roots (where the contour intersects 0 as I have written your function). A benefit is that the contour function returns the (x,y) coordinates of the roots as well.

k=50;
x = 0.01:1:10;
[~,n] = size(x);
y = linspace(0, 50, 150);

eqn = @(y,x) log(y)+log(((k-1-x)./(k-1-y.*x)).^k) - log(4);

[X,Y] = meshgrid(x,y);
Z = eqn(X,Y);

figure(1)
contour(X, Y, Z, [0 0])
grid on
set(gca, 'YLim', [-5  50])
xlabel('X')
ylabel('Y')

See the documentation on the contour (link) function for details. You can get the (x,y) coordinates of that contour by getting the first output from contour, for example:

C = contour(X, Y, Z, [0 0]);

Xv = C(1,2:end);
Yv = C(2,2:end);

figure(2)
semilogx(Xv, 10*log10(Yv))
grid

You can then convert the ‘y’-cordinates to dB, and plot them as s function of the ‘x’-coordinates. See the documentation on the ContourMatrix (link) property for details and a description of how the data are stored, and how to access them. I already did that here.

I defer to you to interpret the results, since I have no idea what you are doing.

Experiment to get the results you want.

Answer 2

[Walter Roberson]

For positive integer k, there are k roots of the equation, most of them imaginary. In the range of x you are interested in, there are two real roots, both of which are positive.

x = 0.01:1:10;
syms X y;
k=50;
eqn = log(y)+log(((k-1-X)/(k-1-y*X))^k) == log(sym(4));
sol = solve(eqn, y);
z = subs(sol(1:2), X, x);
subplot(1,2,1)
semilogy(x, z(1,:))
title('lower branch')
subplot(1,2,2)
semilogy(x, z(2,:))
title('upper branch')