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

February 22, 2026

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.

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

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

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

Plug in: d ≈ 0.6931 / 0.09531 ≈ 7.27

So, (1.10)^d > 2 when d > 7.27.

Step 5: Final Answer

Since (1.10)^d > 2 is true for all d greater than approximately 7.27, the solution to the original inequality is: d > 7.27

This means your growth of 10% per time unit exceeds double the base value when d exceeds about 7.27 time units.

Summary

Start with 120 × (1.10)^d > 240
Simplify to (1.10)^d > 2
Solve using logarithms: d = ln(2) / ln(1.10)
Numerical solution: d > 7.27

Why This Matters Practically

Understanding when exponential growth doubles helps in:

Calculating how long to double investments with 10% annual interest
Modeling time required for population or biological systems to grow
Solving real-world problems in finance, biology, and physics

Use this method anytime you encounter exponential inequalities — from simple growth models to advanced scientific calculations.

For deeper insights, explore logarithmic functions and logarithmic identities to master exponential equations effortlessly!

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