To solve the equation:

$17 \cosh(x) + 18 \sinh(x) = 1$

Step 1: Express the hyperbolic functions in terms of exponentials.

Recall that:

• $\sinh(x) = \frac{e^x - e^{-x}}{2}$
• $\cosh(x) = \frac{e^x + e^{-x}}{2}$

Substituting these into the equation:

$17 \left(\frac{e^x + e^{-x}}{2}\right) + 18 \left(\frac{e^x - e^{-x}}{2}\right) = 1$

Step 2: Simplify the equation.

Multiplying through by 2 to eliminate the fractions:

$17(e^x + e^{-x}) + 18(e^x - e^{-x}) = 2$

Expanding:

$17e^x + 17e^{-x} + 18e^x - 18e^{-x} = 2$

Combining like terms:

$(17e^x + 18e^x) + (17e^{-x} - 18e^{-x}) = 2$

Simplifying further:

$35e^x - e^{-x} = 2$

Step 3: Solve the equation.

This equation is also transcendental, so solving for $e^x$ requires either substitution or a numerical method. Let's solve it numerically to determine the real values of $x$.

I'll calculate this for you.The solutions for the equation $17 \cosh(x) + 18 \sinh(x) = 1$ include:

• $x = -\log(7) + i\pi$ (a complex solution)
• $x = -\log(5)$ (a real solution)

Therefore, the real solution is:

$x = -\log(5)$

This represents the only real solution to the equation.

Would you like further clarification or numerical approximation of this result?

Here are five related questions to explore:

1. How can logarithms simplify solving exponential equations?
2. Why do hyperbolic equations often have complex solutions?
3. What are the applications of hyperbolic functions in physics?
4. How do real and complex solutions differ in their interpretation?
5. What is the relationship between the logarithm function and exponential growth?

Tip: When solving transcendental equations, always check for both real and complex solutions, as the behavior of hyperbolic functions can lead to both types of solutions.