How to Solve 3^x + 3^x = 350: Complete Step-by-Step Guide

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Exponential equations can seem intimidating at first glance, but with the right approach, they become manageable mathematical puzzles. Today, we'll tackle the equation 3^x + 3^x = 350 and learn valuable techniques that apply to similar exponential problems.

Whether you're a high school student preparing for exams, a college student reviewing algebra, or someone refreshing their math skills, this guide will walk you through every step of the solution process.

Understanding the Problem: 3^x + 3^x = 350

Before diving into the solution, let's examine what we're working with:

  • Equation type: Exponential equation with identical terms
  • Variables: One unknown variable (x)
  • Complexity level: Intermediate algebra
  • Key insight: The equation contains two identical exponential terms

Step-by-Step Solution Method

Step 1: Simplify the Left Side

The first crucial observation is that we have two identical terms on the left side:

3^x + 3^x = 350

Since both terms are identical, we can factor: 2 × 3^x = 350

This simplification transforms our equation into a much more manageable form.

Step 2: Isolate the Exponential Term

Now we divide both sides by 2:

3^x = 350 ÷ 2

3^x = 175

We've successfully isolated the exponential term, bringing us closer to finding x.

Step 3: Apply Logarithms

To solve for x, we need to "bring down" the exponent. We do this using logarithms:

log(3^x) = log(175)

Using the logarithm power rule, log(a^b) = b × log(a):

x × log(3) = log(175)

Step 4: Solve for x

Finally, we isolate x by dividing both sides by log(3):

x = log(175) ÷ log(3)

Step 5: Calculate the Numerical Answer

Using a calculator (or logarithm properties):

  • log(175) ≈ 2.2430
  • log(3) ≈ 0.4771

Therefore: x = 2.2430 ÷ 0.4771 ≈ 4.70

Verification: Checking Our Answer

Let's verify our solution by substituting x ≈ 4.70 back into the original equation:

  • 3^4.70 ≈ 175
  • 3^4.70 + 3^4.70 ≈ 175 + 175 = 350 ✓

Our answer checks out!

Alternative Solution Methods

Method 2: Using Natural Logarithms

We could also solve using natural logarithms (ln):

x = ln(175) ÷ ln(3)

This gives the same result: x ≈ 4.70

Method 3: Graphical Approach

You can visualize this problem by graphing:

  • y = 2 × 3^x
  • y = 350

The intersection point gives you the solution.

Common Mistakes to Avoid

When solving exponential equations like 3^x + 3^x = 350, students often make these errors:

  1. Forgetting to combine like terms: Not recognizing that 3^x + 3^x = 2 × 3^x
  2. Logarithm confusion: Mixing up different logarithm bases
  3. Calculator errors: Using degrees instead of the natural logarithm setting
  4. Verification oversight: Not checking the answer in the original equation

Practice Problems

Try solving these similar equations:

  1. 2^x + 2^x = 32
  2. 5^x + 5^x = 250
  3. 4^x + 4^x + 4^x = 192

Answers: x = 4, x ≈ 2.86, x = 2

 

Conclusion

Solving 3^x + 3^x = 350 demonstrates the power of recognizing patterns in mathematics. By simplifying to 2 × 3^x = 350, then 3^x = 175, we transformed a seemingly complex equation into a straightforward logarithmic problem.

The solution x ≈ 4.70 shows that mathematical precision combined with logical reasoning can tackle even challenging exponential equations. Remember to always verify your answers and consider multiple solution approaches to build confidence in your mathematical problem-solving skills.

 

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