Let's solve the problem step by step.

Given:

The equation is $5^x = 2^x + 1$.

(a) Exact Solution in terms of Logarithms:

1. Rearrange the equation to isolate exponential terms: $5^x - 2^x = 1$

2. Taking the natural logarithm on both sides: $\ln(5^x) = \ln(2^x + 1)$

3. Applying the logarithmic identity $\ln(a^b) = b\ln(a)$: $x \cdot \ln(5) = \ln(2^x + 1)$

However, this form doesn’t directly yield a simple expression. Thus, the equation cannot be solved explicitly using elementary logarithms. Instead, numerical methods or graphing are typically required.

(b) Approximate Solution using a Calculator:

We need to find the value of $x$ that satisfies $5^x = 2^x + 1$ to six decimal places.

We can solve this numerically. Let me do this calculation.The approximate solution to the equation $5^x = 2^x + 1$ is:

$x \approx 0.563896$

Summary:

• (a) The exact solution in terms of logarithms cannot be simplified directly.
• (b) The numerical approximation is $x = 0.563896$, rounded to six decimal places.

Would you like to see more details, or do you have any questions?

Related Questions:

1. How do you solve equations involving different bases using logarithms?
2. What are the properties of exponential functions used here?
3. How can we use graphing techniques to find approximate solutions?
4. What is the significance of the initial guess in numerical methods like $fsolve$?
5. Can this problem be solved using other numerical methods like Newton’s method?

Tip:

Always check the validity of an approximate solution by substituting it back into the original equation.