The image shows a mathematical equation that needs to be solved. The equation is:

$10^x + 5 = 3^{2x}$

Solution:

To solve the equation $10^x + 5 = 3^{2x}$, we can try finding solutions through various methods. Here’s a step-by-step approach:

1. Try substitution or inspection:

   • Substitute simple values of $x$ to see if we get an equality.

2. Numerical Solution:

   • This is not easily factorizable, and solving this algebraically would require advanced methods like logarithms or using numerical methods.

Let's test a few values of $x$:

• If $x = 0$:
  $10^0 + 5 = 1 + 5 = 6$
  $3^{2 \times 0} = 3^0 = 1$
  $6 \neq 1$, so $x = 0$ is not a solution.

• If $x = 1$:
  $10^1 + 5 = 10 + 5 = 15$
  $3^{2 \times 1} = 3^2 = 9$
  $15 \neq 9$, so $x = 1$ is not a solution.

• If $x = 2$:
  $10^2 + 5 = 100 + 5 = 105$
  $3^{2 \times 2} = 3^4 = 81$
  $105 \neq 81$, so $x = 2$ is not a solution.

Since simple substitutions don't yield solutions, we would normally use a numerical method or graph both sides of the equation to approximate where the two functions intersect. If we graph $10^x + 5$ and $3^{2x}$, the point where the graphs intersect will give the solution.

However, based on the context, this problem might be more suitable to solve using numerical methods like Newton's method or by using a calculator to find an approximate solution.

Would you like to proceed with a more detailed explanation of finding the solution numerically or graphically?

Follow-up Questions:

1. What are the basic properties of exponential functions that can be used to solve equations like this one?
2. How does the graph of $10^x + 5$ compare to the graph of $3^{2x}$?
3. Why is substitution not always an effective method for solving equations with exponents?
4. What numerical methods can be used to solve complex exponential equations?
5. How does Newton's method work in finding the roots of such equations?

Tip: Always check the behavior of exponential functions for different ranges of $x$, as they can grow or decay rapidly, giving you insights into the solution.