Logarithm question

[nicholaskong100]

I'm not entirely sure how 3 as an exponent became 1/3? Is there a name of a formula or a step by step on how to arrive to this solution?

 

[Pedja]

$$\log_{a^n}b=\frac{1}{\log_ba^n}=\frac{1}{n \cdot \log_ba}=\frac{1}{n} \cdot \frac{1}{\log_ba}=\frac{1}{n}\cdot \log_ab$$

 

[Otis]

Is there a name of a formula

Hi Nicholas. There's no formula. We use what we know about logs and exponents (i.e., definitions and properties -- for writing/manipulating expressions).

Do you know how to switch back and forth between exponential form and logarithmic form? That could be one approach.

Let the right-hand side be y.

$$\log_{2^3}(x) = y$$
Switch to exponential form, and then solve for the variable in the exponent.

?

 

[pka]

The standard change of base is $\log_b(a)=\dfrac{\log(a)}{\log(b)}$

 

[HallsofIvy]

Do you understand that $y= log_a(x)$ is the same as $x= a^y$?
(That's the "exponential form" Otis mentioned. Basically, $a^x$ and $log_a(x)$ are "inverse functions".)

Saying that $y= log_{2^3}(x)$ is the same as saying that $x= (2^3)^y= 2^{3y}$.
Going back, then, $3y= log_2(x)$ or $y=\frac{1}{3}log_2(x)$.

 

[Otis]

Saying that $y= log_{2^3}(x)$ is the same as saying that $x= (2^3)^y= 2^{3y}$.

Going back, then, $3y= log_2(x)$ ...

Thanks for following up with that. I like this approach because, even if a student doesn't realize that the definition can be used with exponential form to go back directly, they ought to possess prior experience of solving for y by taking logs and using the property to get y out of the exponent position.

 

[nicholaskong100]

Do you understand that $y= log_a(x)$ is the same as $x= a^y$?
(That's the "exponential form" Otis mentioned. Basically, $a^x$ and $log_a(x)$ are "inverse functions".)

Saying that $y= log_{2^3}(x)$ is the same as saying that $x= (2^3)^y= 2^{3y}$.
Going back, then, $3y= log_2(x)$ or $y=\frac{1}{3}log_2(x)$.

Thanks, oh. I didn't know. Now, I understand it now.

 

[nicholaskong100]

Hi Nicholas. There's no formula. We use what we know about logs and exponents (i.e., definitions and properties -- for writing/manipulating expressions).

Do you know how to switch back and forth between exponential form and logarithmic form? That could be one approach.

Let the right-hand side be y.

$$\log_{2^3}(x) = y$$
Switch to exponential form, and then solve for the variable in the exponent.

?

Thanks I find out how to turn log into exponent and it's much easier to understand now. Thank you for the keywords, Otis.