Find the areas between the curves y=2(log_2(x))/x and y=2(log_4(x))/x and the x-axis from x=1 to x=e. What is the ratio of the larger area to the smaller?

Verified step by step guidance

1. First, rewrite the logarithmic expressions in terms of natural logarithms to simplify the integrals. Recall that \( \log_a(x) = \frac{\ln(x)}{\ln(a)} \). So, express \( y = \frac{2 \log_2(x)}{x} \) as \( y = \frac{2}{x} \cdot \frac{\ln(x)}{\ln(2)} \) and \( y = \frac{2 \log_4(x)}{x} \) as \( y = \frac{2}{x} \cdot \frac{\ln(x)}{\ln(4)} \).
2. Set up the definite integrals for the areas between each curve and the x-axis from \( x=1 \) to \( x=e \). The area under each curve is given by \( A = \int_1^e y \, dx \). So, write the integrals as \( A_1 = \int_1^e \frac{2}{x} \cdot \frac{\ln(x)}{\ln(2)} \, dx \) and \( A_2 = \int_1^e \frac{2}{x} \cdot \frac{\ln(x)}{\ln(4)} \, dx \).
3. Factor out constants from the integrals to simplify. For example, \( A_1 = \frac{2}{\ln(2)} \int_1^e \frac{\ln(x)}{x} \, dx \) and similarly for \( A_2 \).
4. Evaluate the integral \( \int_1^e \frac{\ln(x)}{x} \, dx \). Use substitution or recall that \( \int \frac{\ln(x)}{x} \, dx = \frac{(\ln(x))^2}{2} + C \). Apply the limits from 1 to \( e \) to find the definite integral value.
5. Calculate the ratio of the larger area to the smaller area by dividing the two expressions \( A_1 \) and \( A_2 \) obtained after integration. Simplify the ratio using properties of logarithms, especially noting that \( \ln(4) = 2 \ln(2) \).

Key Concepts

Logarithmic Functions and Change of Base

Understanding logarithmic functions, especially with different bases, is crucial. The change of base formula, log_b(x) = log_k(x) / log_k(b), allows rewriting logs in a common base to simplify expressions and integrals.

Definite Integration for Area Calculation

Calculating the area under a curve between two points involves evaluating the definite integral of the function over that interval. This process sums infinitesimal slices to find the total area bounded by the curve and the x-axis.

Comparing Areas and Ratios

After finding the areas under each curve, comparing their sizes by forming a ratio helps quantify their relative magnitudes. This involves dividing the larger area by the smaller to express how many times one area exceeds the other.