Better: solve $ 64 \times (1.125)^t > 95 $ → $ t = \log_{1.125}(95/64) \approx \log(1.484375)/\log(1.125) \approx 0.17116 / 0.051161 \approx 3.348 $
["Solving the Exponential Inequality $ 64 \ imes (1.125)^t > 95 $: A Step-by-Step Guide", "Understanding and solving exponential inequalities is essential in fields like finance, science, and engineering. One such common problem is determining when a quantity grows beyond a threshold using exponential growth:\n[\n64 \ imes (1.125)^t > 95\n]", "This article explains how to solve this inequality step-by-step and analyzes the numerical solution using logarithms.", "---", "### Step 1: Isolate the Exponential Term", "Start by isolating the exponential factor:", "[\n(1.125)^t > \frac{95}{64}\n]", "We calculate the ratio:", "[\n\frac{95}{64} = 1.484375\n]", "So, the inequality becomes:", "[\n(1.125)^t > 1.484375\n]", "---", "### Step 2: Apply Logarithms to Both Sides", "To solve for $ t $, apply the logarithm base $ 1.125 $:", "[\nt > \log_{1.125}(1.484375)\n]", "This is an exact expression: $ t = \log_{1.125}(1.484375) $", "---", "### Step 3: Convert to Common Logarithms", "Using the change of base formula:", "[\n\log_{a}(b) = \frac{\log(b)}{\log(a)}\n]", "Apply it to the expression:", "[\nt = \frac{\log(1.484375)}{\log(1.125)}\n]", "Using base-10 logarithms (common logs), we compute:", "- $ \log(1.484375) \approx 0.17116 $\n- $ \log(1.125) \approx 0.051161 $", "Thus:", "[\nt \approx \frac{0.17116}{0.051161} \approx 3.348\n]", "---", "### Step 4: Mechanical Interpretation of the Result", "The solution tells us that for the influence or value described by $ 64 \cdot (1.125)^t $ to exceed 95, time $ t $ must be greater than approximately 3.348. Since $ t $ represents time or growth periods, this means the threshold is crossed after about 3.35 units (e.g., years, days, or compounding intervals), depending on the context.", "---", "### Why Use Logarithms?", "Because $ 1.125 $ is not a nice whole number, direct arithmetic does not yield $ t $ easily. Logarithms convert multiplicative growth into linear scaling, making it straightforward to solve for $ t $ algebraically.", "---", "### Final Summary", "- The inequality $ 64 \ imes (1.125)^t > 95 $ solves to $ t > \log_{1.125}(1.484375) $\n- Numerically, $ t > \frac{\log(1.484375)}{\log(1.125)} \approx 3.348 $\n- This means the solution starts around $ t \approx 3.35 $ in the growth model", "Mastering such exponential solving techniques empowers problem-solving in compound interest, population growth, radioactive decay, and many other real-world scenarios.", "---", "Keywords for SEO: exponential inequality solution, how to solve $ 64 \ imes (1.125)^t > 95 $, logarithmic approach exponential growth, $ t = \log_{1.125}(95/64) $, numerical solution step-by-step, understanding compound growth with logs."]
