In the figure log₁₀y is plotted against log₁₀x

When y is plotted against x, then the plot in the provided range is

Correct option is D

Given:

The plot of log⁡10y\log_{10} ylog₁₀y versus log⁡10x\log_{10} xlog₁₀x is a straight line with a positive slope $m>0$ and a y-intercept ccc. This indicates a relationship of the form:

log₁₀y = m log₁₀x+c

Exponentiating both sides, we get:

y = 10_c.x_m

This equation describes a power-law relationship between $y$ and $x$.

Concept Used:

1. Power-law relationships in log-log plots:

   • In a log-log plot, if y = 10_c.x_m. the relationship appears as a straight line.
   • The slope of the straight line is $m$, and the intercept is determined by $c$.

2. Given information:

   • The plot of log⁡10y\log_{10} ylog₁₀y vs. log⁡10x\log_{10} xlog_$10$x is a straight line, so the relationship remains linear on a log-log scale.

Analysis of Options:

• Option A: This is a curve, not a straight line, so it is incorrect.
• Option B: This starts curving upwards, inconsistent with a power-law straight line on a log-log scale.
• Option C: This shows a curve on a log-log scale, which does not match the straight-line behavior of a power-law relationship.
• Option D: This is a straight line on a log-log scale, which correctly represents the power-law relationship y = 10_c.x_m.

Answer:

The correct option is D.