Divide both sides by n (n > 0): 0.05 log₂(n) < 0.01n → log₂(n) < 0.2n.

["Solving the Inequality: Divide Both Sides by n (n > 0) – 0.05 log₂(n) < 0.01n → log₂(n) < 0.2n", "Understanding how to manipulate and solve logarithmic inequalities is essential in fields like mathematics, engineering, economics, and data science. One common type of inequality involves dividing both sides by a positive variable, especially when that variable influences growth rates, such as in logarithmic and linear functions.", "In this article, we focus on the key inequality:", "[\n0.05,\log_2(n) < 0.01n \quad \ ext{where } n > 0\n]", "We’ll walk through the steps of dividing both sides by ( n ) (a positive scalar since ( n > 0 )), transforming the inequality, and analyzing the resulting expression involving ( \log_2(n) ) and ( n ).", "---", "### Step 1: Divide both sides by ( n )", "Since ( n > 0 ), dividing both sides of the inequality by ( n ) preserves the direction of the inequality:", "[\n\frac{0.05,\log_2(n)}{n} < 0.01\n]", "Rewriting 0.05 as ( \frac{1}{20} ), we get:", "[\n\frac{\log_2(n)}{20n} < 0.01\n]", "Multiply both sides by 20 to simplify:", "[\n\frac{\log_2(n)}{n} < 0.2\n]", "---", "### Step 2: Reformulate the inequality", "We now have:", "[\n\log_2(n) < 0.2n\n]", "This inequality compares a slowly growing logarithmic function ( \log_2(n) ) to a linearly increasing function ( 0.2n ). The goal is to find the range of positive ( n ) for which this holds.", "---", "### Step 3: Intuitive insight and graphical interpretation", "Plot or analyze the two functions ( f(n) = \log_2(n) ) and ( g(n) = 0.2n ) on the same graph. The curve ( \log_2(n) ) starts near 0 for small ( n ) and grows slowly. The line ( 0.2n ) grows steadily and faster than the logarithm.", "The inequality ( \log_2(n) < 0.2n ) holds initially (for small ( n )), but at some point exceeds it. The boundary point where they intersect defines the solution interval.", "---", "### Step 4: Solve ( \log_2(n) = 0.2n ) numerically", "The equation", "[\n\log_2(n) = 0.2n\n]", "does not yield a closed-form algebraic solution, so we use numerical methods or estimation.", "Rewrite in natural logarithm form:", "[\n\frac{\ln(n)}{\ln(2)} = 0.2n \quad \Rightarrow \quad \ln(n) = 0.2n \ln(2)\n]", "Let ( k = \ln(2) \approx 0.6931 ), so:", "[\n\ln(n) = 0.2k,n \quad \Rightarrow \quad \ln(n) - 0.2k,n = 0\n]", "Define ( h(n) = \ln(n) - 0.2k,n ). We seek the positive root(s) of ( h(n) = 0 ).", "Test values around suspected bounds:", "- At ( n = 1 ): ( \ln(1) - 0.2k(1) = 0 - 0.1386 = -0.1386 ) (negative)\n- At ( n = 2 ): ( \ln(2) - 0.2k \cdot 2 = 0.6931 - 0.1386 \cdot 2 = 0.6931 - 0.2772 = 0.4159 ) (positive)", "So solution is between ( n = 1 ) and ( n = 2 ).", "Try ( n = 1.8 ):", "[\n\ln(1.8) \approx 0.5878,\quad 0.2k \cdot 1.8 \approx 0.1386 \cdot 1.8 = 0.2495 \Rightarrow 0.5878 - 0.2495 = 0.3383 \ (\ ext{positive})\n]", "Try ( n = 1.5 ):", "[\n\ln(1.5) \approx 0.4055,\quad 0.1386 \cdot 1.5 = 0.2079 \Rightarrow 0.4055 - 0.2079 = 0.1976 \ (\ ext{positive})\n]", "Try ( n = 1.3 ):", "[\n\ln(1.3) \approx 0.2624,\quad 0.1386 \cdot 1.3 = 0.1802 \Rightarrow 0.0822 \ (\ ext{positive})\n]", "Try ( n = 1.2 ):", "[\n\ln(1.2) \approx 0.1823,\quad 0.1386 \cdot 1.2 = 0.1663 \Rightarrow 0.1823 - 0.1663 = 0.016 \ (\ ext{positive})\n]", "Try ( n = 1.1 ):", "[\n\ln(1.1) \approx 0.0953,\quad 0.1386 \cdot 1.1 = 0.1525 \Rightarrow 0.0953 - 0.1525 = -0.0572 \ (\ ext{negative})\n]", "So the root lies between ( n = 1.1 ) and ( n = 1.2 ).", "Refine to approximate solution:", "- Try ( n = 1.15 ):\n ( \ln(1.15) \approx 0.1398 ),\n ( 0.1386 \cdot 1.15 \approx 0.1594 \Rightarrow 0.1398 - 0.1594 = -0.0196 ) (negative)", "- Try ( n = 1.18 ):\n ( \ln(1.18) \approx 0.1655 ),\n ( 0.1386 \cdot 1.18 \approx 0.1638 \Rightarrow 0.1655 - 0.1638 = 0.0017 ) (positive)", "So root ≈ 1.175", "Thus, ( \log_2(n) < 0.2n ) holds for ( 0 < n < n_0 ), where ( n_0 \approx 1.175 )", "---", "### Step 5: Final conclusion", "Dividing the original inequality", "[\n0.05 \log_2(n) < 0.01n \quad (n > 0)\n]", "by ( n ) gives:", "[\n\frac{0.05}{n} \log_2(n) < 0.01 \quad \Rightarrow \quad \log_2(n) < 0.2n\n]", "This inequality holds only when ( n < n_0 ), where ( n_0 \approx 1.175 ). Beyond this point, ( \log_2(n) ) grows faster than ( 0.2n ), and the inequality reverses.", "---", "### Practical takeaway", "- Use division by positive ( n ) to simplify inequalities without changing direction.\n- Recognize logarithmic growth lags behind linear growth for larger ( n ).\n- Always verify boundary behavior using numerical approximation or graphical tools.\n- This technique is valuable in modeling scenarios such as cost-per-user growth, population thresholds, or computational complexity comparisons.", "---", "Keywords:\n( \log_2(n) < 0.2n ), divide both sides by n, inequality solving, logarithmic inequality, mathematical analysis, growth rate comparison, inequality transformation, numerical solution logarithmic, exponential boundary.", "---", "References:\n- Logarithmic function growth properties\n- Numerical root approximation techniques\n- Real-valued function comparison analysis", "---", "Understanding how manipulations like dividing by ( n ) affect inequalities equips you with foundational tools to solve complex mathematical expressions and communicate precise mathematical reasoning—key for academic rigor and technical communication."]