### Compare two graphs of logarithm functions

Question Sample Titled 'Compare two graphs of logarithm functions'

The figure shows the graph of ${y}={b}^{{x}}$ and the graph of ${y}={c}^{{x}}$ on the same rectangular coordinate system, where ${b}$ and ${c}$ are positive constants. If a horizontal line ${L}$ cuts the ${y}$-axis, the graph of ${y}={b}^{{x}}$ and the graph of ${y}={c}^{{x}}$ at ${A}$, ${B}$ and ${C}$ respectively, which of the following are true?

${x}$${y}$${O}$${A}$${B}$${C}$${y}={b}^{{x}}$${y}={c}^{{x}}$${L}$

 I. ${b}>{c}$ II. ${b}{c}>{1}$ III. $\dfrac{{{A}{B}}}{{{A}{C}}}={{\log}_{{b}}{c}}$

A
I, II and III
B
I and II only
C
I and III only
D
II and III only

 (I) Observe that for ${x}>{0}$, the graph of ${y}={b}^{{x}}$ lies above the graph of ${y}={c}^{{x}}$  ∴   For ${x}>{0},{b}^{{x}}>{c}^{{x}}$  You may substitute ${x}={1}$ to check ∴  ${b}>{c}$  (I) is true. (II) Observe that when ${x}$ increases, ${y}$ would increase. ∴  ${b}>{1}$ and ${c}>{1}$ . ∴  ${b}{c}>{1}$  (II) is true. (III) Assume the line ${L}$ is ${y}={d}$ . Find the length of ${A}{B}$ and ${A}{C}$ in terms of ${b}$ and ${c}$ first. To find ${A}{B}$ is equivalent to finding the ${x}$-coordinate of point ${B}$ . Substitute ${y}={d}$ into the equation ${y}={b}^{{x}}$ , ${d}={b}^{{x}}$  ${{\log}_{{b}}{d}}={{\log}_{{b}}{\left({b}^{{x}}\right)}}$ Take logarithm on both sides ${x}={{\log}_{{b}}{d}}$ ∴  ${A}{B}={{\log}_{{b}}{d}}$ Similarly, ${A}{C}={{\log}_{{c}}{d}}$ . ∴  $\dfrac{{{A}{B}}}{{{A}{C}}}=\dfrac{{{{\log}_{{b}}{d}}}}{{{{\log}_{{c}}{d}}}}$  $=\dfrac{{{\left(\dfrac{{\log{{d}}}}{{\log{{b}}}}\right)}}}{{{\left(\dfrac{{\log{{d}}}}{{\log{{c}}}}\right)}}}$  Change of base $={\left(\dfrac{{\log{{d}}}}{{\log{{b}}}}\right)}\times{\left(\dfrac{{\log{{c}}}}{{\log{{d}}}}\right)}$  $=\dfrac{{\log{{c}}}}{{\log{{b}}}}$  $={{\log}_{{b}}{c}}$  (III) is true.

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