Solving the Equation 120 × (1.10)^d > 240: A Step-by-Step Explanation

If you’ve ever wondered how to solve exponential inequalities like 120 × (1.10)^d > 240, you’re in the right place. In this article, we’ll break down the process clearly and show you how to solve (1.10)^d > 2, a simplified version of the original inequality, using fundamental mathematical principles.

Understanding the Context

Why This Equation Matters

Exponential functions model real-world phenomena such as compound interest, population growth, and radioactive decay. Understanding how to solve equations of the form a × b^d > c helps in finance, science, and engineering. Our focus here is solving (1.10)^d > 2, a common form that appears when analyzing growth rates.

Step 1: Simplify the Inequality

We solve: 120 × (1.10)^d > 240 → (1.10)^d > 2

Key Insights

Start with the original inequality:
120 × (1.10)^d > 240

Divide both sides by 120:
(1.10)^d > 2

Now we solve this exponential inequality — a key step toward understanding how the base (1.10) grows over time d.

Step 2: Solve the Corresponding Equation

Continue Reading

We require \( n_k \leq 50 \):
7k - 4 \leq 50
7k \leq 54

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Final Thoughts

To isolate the exponent d, first convert the inequality into an equation by changing the “>” to “=”:
(1.10)^d = 2

This helps us find the threshold value of d beyond which the inequality holds.

Step 3: Take the Logarithm of Both Sides

Use logarithms to bring the exponent down:
Take natural logarithm (ln) or common logarithm (log) — either works.
Apply ln:
ln((1.10)^d) = ln(2)

Use the logarithmic identity: ln(a^b) = b·ln(a)
This gives:
d · ln(1.10) = ln(2)

Step 4: Solve for d

Now isolate d:
d = ln(2) / ln(1.10)

Using approximate values:

  • ln(2) ≈ 0.6931
  • ln(1.10) ≈ 0.09531