We solve: 120 × (1.10)^d > 240 → (1.10)^d > 2 - Richter Guitar
Solving the Equation 120 × (1.10)^d > 240: A Step-by-Step Explanation
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
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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
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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
