Solve the equation 13(8^x+2)=52 using common bases. Briefly explain each step
in your process. 

To solve the equation \( 13(8^x + 2) = 52 \) using common bases, we will follow these steps:

### Step 1: Simplify the equation
We start by dividing both sides by 13 to simplify the equation:
\[
8^x + 2 = \frac{52}{13}
\]
\[
8^x + 2 = 4
\]

### Step 2: Isolate the exponential term
Now, subtract 2 from both sides to isolate the term with the exponential expression:
\[
8^x = 4 - 2
\]
\[
8^x = 2
\]

### Step 3: Express both sides as powers of 2
We now express both sides using powers of 2. Notice that:
\[
8 = 2^3 \quad \text{so} \quad 8^x = (2^3)^x = 2^{3x}
\]
And the right-hand side is already in base 2:
\[
2 = 2^1
\]

### Step 4: Set the exponents equal to each other
Since the bases are the same, we can equate the exponents:
\[
3x = 1
\]

### Step 5: Solve for \( x \)
Now, divide both sides by 3 to solve for \( x \):
\[
x = \frac{1}{3}
\]

Thus, the solution to the equation is \( x = \frac{1}{3} \).

---

Would you like more details or have any questions? Here are five related questions to explore:

1. How do you solve equations involving different exponential bases?
2. What are common techniques for handling logarithmic equations?
3. How can you express numbers as powers of smaller primes like 2, 3, or 5?
4. How would the solution change if the equation had a different constant term?
5. How do you solve exponential equations without common bases?

**Tip:** When solving exponential equations, look for ways to express both sides using the same base—it simplifies the process greatly!

Solve the equation 13(8^x + 2) = 52 using common bases. Briefly explain each step in your process.

Step-by-step guide to solving the exponential equation 13(8^x + 2) = 52 using common bases. Learn how to express exponents with the same base, isolate the exponential term, and solve for x. Suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation