MHB Logarithm and Exponent Question

[zekea]

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

 

[mathmari]

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 $$ p^{\frac{q}{m}}=n$$

 

[zekea]

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 ?

 

[mathmari]

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: $$n^m=p^q \Rightarrow \left (n^m\right )^{\frac{1}{m}}=\left (p^q\right )^{\frac{1}{m}} \Rightarrow n^{\frac{m}{m}}=p^{\frac{q}{m}} \Rightarrow n=p^{\frac{q}{m}}$$

 

[MarkFL]

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:

$$c=a^b$$