Logarithm and Exponent Questions: write m log p (n) = q as exponential eqn

[zekea]

I'm confused on these questions.

1. The equation m log p (n) = q can be written in exponential form as..
The answer on the work sheet is p^(q/m)=n but shouldn't it be P^(qm) = n ? According to the power rule? My teacher explained this by writing down for me log p (n) = q / m but I'm confused here

2. The graph of y=3^x is reflected in the line y=x. The equation of the transformed graph is
The answer is y=log 3 (x)
This doesn't make any sense to me either. Shouldn't it be log 3 (y) = x

 

[stapel]

1. The equation m log p (n) = q can be written in exponential form as..
The answer on the work sheet is p^(q/m)=n but shouldn't it be P^(qm) = n ?

Please write out the steps you used to arrive at your answer. You started by applying the power rule to move the "m" inside the log as an exponent. (here) Then you converted the equation, using the basic relationship between logs and exponentials. (here) And... then what?

2. The graph of y=3^x is reflected in the line y=x. The equation of the transformed graph is
The answer is y=log 3 (x)
This doesn't make any sense to me either. Shouldn't it be log 3 (y) = x

If you solve your equation for "y=", you get:

. . . . .\(\displaystyle \log_3(y)\, =\, x\)

. . . . .\(\displaystyle 3^{\log_3(y)}\, =\, 3^x\)

. . . . .\(\displaystyle y\, =\, 3^x\)

Is this any sort of transformation of the original graph?

What did you get, when you graphed the original equation and then did the transformation?

 

[ksdhart2]

For the first problem, it's true that you can run the power rule "in reverse" to arrive at \(\displaystyle log_p(n^m)=q\), but I'm not seeing how you got from there to your proposed answer of \(\displaystyle p^{qm}=n\). I still arrive at the answer given by your teacher. Can you show your steps please? As for the teacher's process, you see how he got to \(\displaystyle log_p(n) = q / m\), right? From there, you have a logarithm (base p) on one side of the equation. How would you "clear" or "undo" that logarithm? As a hint, consider how you might tackle it if it were the natural log (logarithm base e) or log base 10.

For the second question, I'm also not seeing how you arrived at your answer of \(\displaystyle x=log_3(y)\). If you "clear" the logarithm, you simply end up with the equation you started with, \(\displaystyle 3^x=y\). A quick peek at a graphing calculator reveals this to be true, if you're unsure. If you take a curve and reflect it across the line y=x, how can you end up with the same curve you started with? Instead, a way to solve it might to be think about what it means to reflect a function across the line y=x. Perhaps start with a simpler function, say y=3x and see what happens. In particular, note what happens to a generic point on the function. What happens to the point (x, y/3) as it's reflected? What does that suggest the "after reflection" function might be? How can you apply that to the actual problem you were given?

 

[zekea]

1. Okay for question one I arrived at that conclusion based on Khans' video here https://www.youtube.com/watch?v=Pb9V374iOas
Skip to 4:00
Basically according to the power rule you have Log a (c)^d = bd . He brought the d down to the other side.
So in exp form A^(bd) = C^d so shouldn't the answer be p^(mn) = n rather than P^(q/m) = n ?

2. Ahh okay I see what you're saying. Yes there's no transformation lol. But then why do they say that a log function is the inverse of an exp function? I thought expressing an exponent function in log form is already making it inverse..

 

[ksdhart2]

Okay, now I think I see what your thought process was. But there's a few stumbling blocks you're encountering. First, what you're saying is the "power rule" for logarithms is really just a step along the way of Khan proving the power rule. \(\displaystyle Log_a(c^d) = bd\) isn't true in most cases. It's only true here because he defined \(\displaystyle a^b=c\) so that \(\displaystyle log_a(c)=b\). If you continue on to the final step of the proof where he replaces b with its equivalent, you arrive at what is really the power rule: \(\displaystyle log_a(c^d)=d\:log_a(c)\).

The given problem starts with a expression that looks like the right-hand of the above equation, so we can substitute the left-hand side instead. That gives us: \(\displaystyle log_p(n^m)=q\) as I said in my first post. From there, a next logical step might be to "clear" the logarithm, leaving us with: \(\displaystyle p^{log_p\left(n^m\right)}=p^q\) or \(\displaystyle n^m=p^q\). But then this doesn't really seem to be leading anywhere. Personally, I wouldn't suggest using this method. You can get to the correct answer this way, but it'll be a lot more steps than you really need.

As for the second problem, it is true, generally speaking, that a logarithm and an exponential are inverses. And it's also true that reflecting across the line y=x is the same as finding the inverse. But, as you've seen, just writing a logarithm as an exponential won't give you its inverse. Instead, try following the standard steps, as shown here and see what you get.

 

[zekea]

Okay, now I think I see what your thought process was. But there's a few stumbling blocks you're encountering. First, what you're saying is the "power rule" for logarithms is really just a step along the way of Khan proving the power rule. \(\displaystyle Log_a(c^d) = bd\) isn't true in most cases. It's only true here because he defined \(\displaystyle a^b=c\) so that \(\displaystyle log_a(c)=b\). If you continue on to the final step of the proof where he replaces b with its equivalent, you arrive at what is really the power rule: \(\displaystyle log_a(c^d)=d\:log_a(c)\).

The given problem starts with a expression that looks like the right-hand of the above equation, so we can substitute the left-hand side instead. That gives us: \(\displaystyle log_p(n^m)=q\) as I said in my first post. From there, a next logical step might be to "clear" the logarithm, leaving us with: \(\displaystyle p^{log_p\left(n^m\right)}=p^q\) or \(\displaystyle n^m=p^q\). But then this doesn't really seem to be leading anywhere. Personally, I wouldn't suggest using this method. You can get to the correct answer this way, but it'll be a lot more steps than you really need.

As for the second problem, it is true, generally speaking, that a logarithm and an exponential are inverses. And it's also true that reflecting across the line y=x is the same as finding the inverse. But, as you've seen, just writing a logarithm as an exponential won't give you its inverse. Instead, try following the standard steps, as shown here and see what you get.

1. Ok I'm feeling you alot more now. So basically the question I first stated is not the same as what Khan is talking about in his video. Cause that's what really is confusing me. He's proving how to derive the law and showing how they are equal to each other whereas the question is asking you to change that into exponential form which apparently is not the same. Is this correct? Now I'm just a bit confused why they are different things. m log p (n) = qM is the law but when you're actually manipulating it you think of it as just m log p (n) = q so you have q/m instead of M multiplying the q? Now this doesn't look equal to me. I think I'm overthinking this.

2. OK, this one I managed to get 100%. Right so you have to swipe the x and y spot and solve for y. So is this a correct statement then? Changing from exponential to logarithm form is not taking the inverse when you change form, you still have to reverse the position of x and y in order to get the proper inverse?

 

[HallsofIvy]

I'm confused on these questions.

Both of these problems use the use the fact that the logarithm is the inverse of the exponential function: if \(\displaystyle log_p(x)= y\) then \(\displaystyle x= p^y\).

1. The equation m log p (n) = q can be written in exponential form as..
The answer on the work sheet is p^(q/m)=n but shouldn't it be P^(qm) = n ? According to the power rule? My teacher explained this by writing down for me log p (n) = q / m but I'm confused here

I would not, as others here did, start by "taking the m inside the logarithm". I would, as it seems your teacher did, first divide both sides by m: \(\displaystyle log_p(n)= \frac{q}{m}\). Then "invert" to get \(\displaystyle n= p^{q/m}\).

2. The graph of y=3^x is reflected in the line y=x. The equation of the transformed graph is
The answer is y=log 3 (x)
This doesn't make any sense to me either. Shouldn't it be log 3 (y) = x

No, \(\displaystyle log_3(y)= x\) is equivalent to the equation you were given, \(\displaystyle y= 3^x\). "Reflecting in the line y= x" swaps x and y so gives \(\displaystyle x= 3^y\). That is equivalent to \(\displaystyle y= log_3(x)\).