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can we have a graph where y = log base x (5) ? or x^y = 5 ?

When I draw this graph out it has a horizontal asymptote at x = 0 on the right end, and it has values in the negative x-axis from -5 < x < -1

Edit : I'm just picking up graphs of logarithms and was wondering if there can be a graph of y = log base x (5), since I havent seen those examples yet

Llama.new
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1 Answers1

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$$x^y=5 \\ y\cdot\ln(x)=\ln(5) \\ y=\dfrac{\ln(5)}{\ln(x)}=\dfrac{\text{constant}}{\ln(x)} $$

This function is easily analyzed by derivatives and graphical methods. It has a vertical asymptote at $x=1$ and approaches $0$ as $x\to0$. Rough graph: Rough graph of y The derivative of the function approaches $-\infty$ as $x\to 0$. Actual graph (from WolframAlpha): Actual graph of y Note the tendency of graph to become vertical near $x=0$.

FreezingFire
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  • ah thanks! and as the value increases , lets say from 5 to 10, the two slopes of the graph deviate further and further away from the point (1,0)? Neat property – Llama.new Sep 30 '16 at 03:14
  • @BrotundWasser What do you mean by "the two slopes of the graph"? – FreezingFire Sep 30 '16 at 11:49
  • I meant the curved portion of the graph. that means the curved section moves away from the point (1,0). As the value increases to 10, when y = 1, x must be 10 (further away) – Llama.new Sep 30 '16 at 14:30
  • @BrotundWasser Oh okay, I got you now! Yes it does. The graph "moves" both up, and right. But note that, in both your cases, $y$ never becomes zero, even though it does get closer and closer! So graph doesn't really move, it somehow "stretches" upward and rightward. Had it moved, it would mean $y$ would never come close to zero, but it does. Keep observing and thinking like this! It opens your mind and helps a lot in learning! :) – FreezingFire Sep 30 '16 at 15:34