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.

Discover how to solve the equation z = 4x + log x + log y by isolating y and applying logarithmic properties. Learn how to express y in terms of z and x using exponentiation. Suitable for high school students in Grades 10-12.

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