We solve: 50 × (1.08)^h > 80 → (1.08)^h > 1.6

["Solving the Exponential Inequality: How to Solve 50 × (1.08)^h > 80", "Understanding and solving exponential inequalities is a key skill in algebra and real-world applications such as finance, population growth, and compound interest. One common type of problem students and professionals encounter is:", "> 50 × (1.08)^h > 80", "This inequality describes growth over time—often used to model compound interest, investment returns, or population increase. In this article, we’ll walk step-by-step through solving this inequality:\n50 × (1.08)ʰ > 80, and explain how to interpret the solution.", "---", "### Step 1: Isolate the Exponential Term", "Start by dividing both sides of the inequality by 50 to simplify:", "[\n(1.08)^h > \frac{80}{50} = 1.6\n]", "Now the inequality becomes:", "[\n(1.08)^h > 1.6\n]", "This is the form we are solving: an exponential expression greater than a number.", "---", "### Step 2: Apply Logarithms to Both Sides", "Since the variable h is in the exponent, the natural approach is to use logarithms. Take the natural log (ln) of both sides:", "[\n\ln\left((1.08)^h\right) > \ln(1.6)\n]", "Using the logarithmic power rule, (\ln(a^b) = b \ln(a)), simplify the left side:", "[\nh \cdot \ln(1.08) > \ln(1.6)\n]", "---", "### Step 3: Solve for h", "Now divide both sides by (\ln(1.08)), but first check the sign of (\ln(1.08)):", "- Since (1.08 > 1), (\ln(1.08) > 0), so dividing does not reverse the inequality:", "[\nh > \frac{\ln(1.6)}{\ln(1.08)}\n]", "Now compute the right-hand side numerically:", "- (\ln(1.6) \approx 0.4700)\n- (\ln(1.08) \approx 0.07696)", "So:", "[\nh > \frac{0.4700}{0.07696} \approx 6.11\n]", "---", "### Step 4: Final Answer", "Thus, the solution to the inequality 50 × (1.08)^h > 80 is:", "[\nh > \frac{\ln(1.6)}{\ln(1.08)} \approx 6.11\n]", "---", "### Interpretation and Applications", "This result means after approximately 6.11 units of time (h), the value (50 \ imes (1.08)^h) exceeds 80. Since h often represents time or growth cycles, this tells us that investment or growth surpasses the threshold after just under 7 full periods.", "For example, if h represents years with an 8% annual growth rate, you can use this to determine when your investment doubles or reaches a target profit.", "---", "### Summary", "- Isolate the exponential term\n- Use logarithms to bring down the exponent\n- Solve for h carefully preserving inequality direction (positive log base)\n- Express the final answer clearly, including numeric approximation", "---", "### Why This Matters", "Understanding how to solve exponential inequalities helps in:", "- Calculating compound interest periods\n- Predicting population or business growth trends\n- Modeling radioactive decay or technology adoption curves", "Mastering this technique empowers you to tackle complex real-world problems with confidence.", "---", "Keywords: exponential inequality, solve 50 × (1.08)^h > 80, logarithmic steps, compound interest inequality, math problem solving, exponential growth model, how to solve exponential inequality", "---", "See more:\n- How to graph exponential inequalities\n- Step-by-step logarithmic inequality solving\n- Applications of exponential functions in finance"]