MHB Logarithm and Exponent Question

AI Thread Summary
The discussion revolves around the confusion regarding the conversion of the logarithmic equation m log p(n) = q into its exponential form. The worksheet states the correct conversion is p^(q/m) = n, but there is a belief that it should be p^(qm) = n based on the power rule of logarithms. The participant references a Khan Academy video to support their reasoning, questioning the application of the power rule. Ultimately, they reiterate the teacher's explanation, which aligns with the worksheet's answer, but remain puzzled about the power rule's implications. The conversation highlights the complexities of understanding logarithmic and exponential relationships.
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I'm confused on this question.

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
 
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zekea said:
I'm confused on this question.

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

The definition of the logarithm function id the following:
If $b$ is any number such that $b>0$ and $b\neq 1$ and $x>0$ then,
$$y=\log_b x \ \ \text{ is equivalent to } \ \ b^y=x$$ We have the the equation $q=m\log_p n$.

Dividing both sides by $m$ we get $$\frac{q}{m}=\frac{m\log_p n}{m} \Rightarrow \frac{q}{m}=\log_p n$$

Therefore from the definition for $y=\frac{q}{m}$, $b=p$ and $x=n$ we get $$ p^{\frac{q}{m}}=n$$
 
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Okay this is what my teacher did but something is confusing me.

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 ?
 
Using that rule we have the following:
$$q=m\log_p n\Rightarrow q=\log_p n^m$$

Then from the definition we get $p^q=n^m$.

To solve for $n$ we do the following: $$n^m=p^q \Rightarrow \left (n^m\right )^{\frac{1}{m}}=\left (p^q\right )^{\frac{1}{m}} \Rightarrow n^{\frac{m}{m}}=p^{\frac{q}{m}} \Rightarrow n=p^{\frac{q}{m}}$$
 
zekea said:
Okay this is what my teacher did but something is confusing me.

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 ?

If I was given:

$$\log_a\left(c^d\right)=bd$$

I would first use the identity $\log_a\left(b^c\right)=c\cdot\log_a(b)$ to write:

$$d\cdot\log_a\left(c\right)=bd$$

Next, divide through by $d$:

$$\log_a\left(c\right)=b$$

Finally, convert from logarithmic to exponential form:

$$c=a^b$$
 
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