LN prob: circle's radius is LN(a^3)^(1/2), circumference LN(b^{2pi}); find log_a(b)

[bobrossu]

Problem:

A circle has a radius of LN(a³)^(1/2) and a circumference of LN(b^2pi). Evaluate log_ab.

LN=natural log

Not too familiar with logs/natural logs and their rules.

Steps would be helpful.

 

[mmm4444bot]

Is this exercise from a math course that you're currently taking?

As posted, there are infinite circles which satisfy the given information. Therefore, it's not possible to evaluate log_a(b) to a single value.

 

[mmm4444bot]

A circle has a radius of LN(a³)^(1/2) and a circumference of LN(b^2pi).

Evaluate log_ab.

I returned because it occurred to me that improper function notation may be the issue, here.

Your typing (highlighted above in color) shows a power; the base in this power is a natural log and the exponent is 1/2.

Raising to the 1/2 is the same as taking the square root.

In other words, the function notation LN(a³)^(1/2) means \(\displaystyle \sqrt{LN(a^3)}\). But, this leads to infinite values for log_a(b).

If a^3 is raised to the 1/2 first, and we take the natural log of the result, then we have LN([a^3]^[1/2]), and this expression for the radius leads to only one value for log_a(b).

Regardless of what you were actually given, we should assume that the circle's radius is LN([a^3]^[1/2]), instead of what you posted.

---

First, I would begin by using the Change-of-Base formula, to express log_a(b) in terms of ln(a) and ln(b).

The Change-of-Base formula is log_u(v) = ln(v) / ln(u).

I would continue, by using a property of exponents to simplify the expression [a^3]^[1/2].

The property is: (c^m)^n = c^(m*n)

Next, I would use a property of logarithms to rewrite each of the expressions for radius and circumference, by "moving the exponent out in front" of the logarithm.

The property for "moving the exponent out in front" is: ln(z^k) = k * ln(z)

Then, I would substitute these rewritten expressions for radius and circumference into the formula for the circumference of a circle.

Solve the resulting equation for ln(b)/ln(a). :cool:

 

[bobrossu]

I returned because it occurred to me that improper function notation may be the issue, here.

Your typing (highlighted above in color) shows a power; the base in this power is a natural log and the exponent is 1/2.

Raising to the 1/2 is the same as taking the square root.

In other words, the function notation LN(a³)^(1/2) means \(\displaystyle \sqrt{LN(a^3)}\). But, this leads to infinite values for log_a(b).

If a^3 is raised to the 1/2 first, and we take the natural log of the result, then we have LN([a^3]^[1/2]), and this expression for the radius leads to only one value for log_a(b).

Regardless of what you were actually given, we should assume that the circle's radius is LN([a^3]^[1/2]), instead of what you posted.

---

First, I would begin by using the Change-of-Base formula, to express log_a(b) in terms of ln(a) and ln(b).

The Change-of-Base formula is log_u(v) = ln(v) / ln(u).

I would continue, by using a property of exponents to simplify the expression [a^3]^[1/2].

The property is: (c^m)^n = c^(m*n)

Next, I would use a property of logarithms to rewrite each of the expressions for radius and circumference, by "moving the exponent out in front" of the logarithm.

The property for "moving the exponent out in front" is: ln(z^k) = k * ln(z)

Then, I would substitute these rewritten expressions for radius and circumference into the formula for the circumference of a circle.

Solve the resulting equation for ln(b)/ln(a). :cool:

So... If I got 1.5*ln(a) for the radius, how to I square it in the formula 2pi(r)² ... I got 2pi*ln(b) for the circumference ------> 2pi(r)²​=2pi*ln(b)... and canceled the 2pi on both sides to get (r)²​=ln(b).

 

[mmm4444bot]

… I got 1.5*ln(a) for the radius … I got 2pi*ln(b) for the circumference …

Good work.

… how [do] I square [the radius] in the formula 2pi(r)² …

You're mixing together parts of the formulas for a circle's circumference and area.

Circumference = 2*Pi*r

Area = Pi*r^2

 

[bobrossu]

Good work.


You're mixing together parts of the formulas for a circle's circumference and area.

Circumference = 2*Pi*r

Area = Pi*r^2

Oh okay. Thanks.

I got log_a​b=1.5