The given equation is:

$z = 4x + \log x + \log y$

Here, $z$ is expressed in terms of $x$ and $y$, where $\log x$ and $\log y$ refer to logarithms (usually assumed to be base 10 unless specified otherwise).

Rearranging or Manipulating the Equation:

1. To isolate $y$, we can subtract $4x + \log x$ from both sides: $z - 4x - \log x = \log y$
2. Exponentiating both sides to eliminate the logarithm: $y = 10^{z - 4x - \log x}$

Thus, $y$ can be expressed as: $y = 10^{z - 4x - \log x}$

Would you like more details or have any specific questions?

Related Questions:

1. How would the equation change if natural logarithms (ln) were used instead of base 10 logarithms?
2. Can you find the partial derivative of $z$ with respect to $x$?
3. What happens to the equation when $z$ is held constant and we differentiate with respect to $x$?
4. How does the value of $y$ change when $z$ is increased by a constant?
5. How can this equation be graphed in a 3D plot with respect to $x$, $y$, and $z$?

Tip:

When dealing with logarithmic equations, always ensure that the argument of the logarithm (here $x$ and $y$) is positive, as logarithms are undefined for non-positive values.