Consider the following.
35x − 1 = 4
(a) Find the exact solution of the exponential equation in terms of logarithms.
x = 
  
Correct: Your answer is correct.


(b) Use a calculator to find an approximation to the solution rounded to six decimal places.
x = 
0.452348
 
Incorrect: Your answer is incorrect.

Let's solve the exponential equation \( 35^x - 1 = 4 \) step by step.

### (a) Exact Solution in Terms of Logarithms

1. Start with the equation:
   \[
   35^x - 1 = 4
   \]

2. Add 1 to both sides:
   \[
   35^x = 5
   \]

3. Take the natural logarithm of both sides:
   \[
   \ln(35^x) = \ln(5)
   \]

4. Use the logarithmic property \( \ln(a^b) = b \ln(a) \) to simplify:
   \[
   x \ln(35) = \ln(5)
   \]

5. Solve for \( x \):
   \[
   x = \frac{\ln(5)}{\ln(35)}
   \]
   This is the exact solution in terms of logarithms.

### (b) Approximation to Six Decimal Places

To find the approximate value, we'll use a calculator to compute \( \frac{\ln(5)}{\ln(35)} \).

1. Calculate \( \ln(5) \approx 1.6094379 \).
2. Calculate \( \ln(35) \approx 3.5553481 \).
3. Now, divide:
   \[
   x \approx \frac{1.6094379}{3.5553481} \approx 0.452414
   \]

So, the approximate solution to six decimal places is:
\[
x \approx 0.452414
\]

The previous approximation of \( 0.452348 \) was incorrect because it was rounded slightly off. The correct value should be \( 0.452414 \).

---

Do you want more details or have any questions about this process?

Here are 5 related questions:
1. How do logarithmic properties simplify equations involving exponents?
2. Why is it important to use natural logarithms instead of common logarithms?
3. What are other ways to solve exponential equations besides using logarithms?
4. Can the equation \( 35^x - 1 = 4 \) have multiple solutions?
5. How would rounding errors affect the solution when using a calculator?

**Tip:** When solving logarithmic problems, always verify calculator entries carefully to avoid rounding errors.

Consider the following. 35^x − 1 = 4 (a) Find the exact solution of the exponential equation in terms of logarithms. (b) Use a calculator to find an approximation to the solution rounded to six decimal places.

Learn how to solve the exponential equation 35^x - 1 = 4 step-by-step, both in terms of logarithms and using a calculator to find an approximation. This problem covers logarithmic properties and the change of base theorem. Ideal for students in Grades 10-12.

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