Solve: 1000 × (0.98)^n < 700 → (0.98)^n < 0.7
["# Solving the Exponential Inequality: 1000 × (0.98)^n < 700", "Mathematical inequalities play a crucial role in various real-world applications, from finance to science, where exponential decay models measurable phenomena like depreciation, radioactive decay, or population decline. One common challenge is solving exponential inequalities of the form:", "1000 × (0.98)^n < 700", "In this article, we’ll walk step-by-step through how to solve this inequality, explain the reasoning behind each transformation, and describe how to interpret the solution practically.", "---", "## Step-by-step Solution", "### Step 1: Isolate the Exponential Term", "Start by dividing both sides of the inequality by 1000 to simplify:", "[\n(0.98)^n < \frac{700}{1000}\n]", "[\n(0.98)^n < 0.7\n]", "This step is valid since dividing by a positive constant preserves the inequality direction.", "---", "### Step 2: Apply Logarithms to Both Sides", "To solve for the exponent ( n ), apply the natural logarithm (ln) to both sides:", "[\n\ln\left((0.98)^n\right) < \ln(0.7)\n]", "Using the logarithmic power rule, ( \ln(a^b) = b \ln a ), rewrite the left-hand side:", "[\nn \cdot \ln(0.98) < \ln(0.7)\n]", "---", "### Step 3: Solve for ( n )", "Now divide both sides by ( \ln(0.98) ). Important: Since ( 0.98 < 1 ), its natural logarithm is negative. When dividing by a negative number, the inequality reverses:", "[\nn > \frac{\ln(0.7)}{\ln(0.98)}\n]", "---", "### Step 4: Compute Numerical Values", "Calculate:", "- ( \ln(0.7) \approx -0.3567 )\n- ( \ln(0.98) \approx -0.0202 )", "[\nn > \frac{-0.3567}{-0.0202} \approx 17.65\n]", "---", "## Final Answer", "The inequality ( 1000 \ imes (0.98)^n < 700 ) holds when:", "[\nn > 17.65\n]", "Since ( n ) typically represents a number of units (like time or iterations), the smallest integer value satisfying the inequality is ( n = 18 ).", "---", "## Practical Interpretation", "This inequality models situations where an initial value of 1000 decays by 2% each period (reflected by the base 0.98). The result tells us that after approximately 18 periods, the quantity drops below 700.", "For example, in financial contexts, this could represent investment depreciation or savings growth slowed by a consistent rate.", "---", "## Summary", "- Start by isolating the exponential term.\n- Use logarithms to bring the exponent down.\n- Remember the sign flip when dividing by a negative logarithm base.\n- Convert to decimal and compute numerically for clear results.\n- Interpret the smallest integer satisfying the inequality for practical use.", "Solving exponential inequalities is a fundamental skill empowering precise modeling and forecasting in mathematics, science, and finance.", "---", "Keywords: exponential inequality, solve 1000(0.98)^n < 700, logarithmic inequality, exponential decay, n > log base 0.98 of 0.7, solve (0.98)^n < 0.7, decay model, real-world applications."]
