Kepler's third law of planetary motion relates P, the period of a planet's orbit, to R, the planet's mean distance from the sun, through the equation log P = 1/2 (log k + 3 log R), where k is a constant.
a) Rewrite the formula as a single logarithm.
b) Rewrite the formula in exponential form.
c) Rewrite the formula, solving for P.
log P = 1/2 log k + 1/2 * 3logR
= log(√k) + log√(R^3)
= log√(kR^3)
P = √(kR^3)
a) To rewrite the formula as a single logarithm, we can use the properties of logarithms.
Given the equation: log P = 1/2 (log k + 3 log R)
We can simplify by applying the exponent rules of logarithms:
log P = 1/2 log(k) + 3/2 log(R)
Now, we can use the property: log(a) + log(b) = log(ab)
log P = log(k^(1/2)) + log(R^(3/2))
Finally, combining the logarithms with the same base, we get:
log P = log((k^(1/2))(R^(3/2)))
Therefore, the formula rewritten as a single logarithm is:
log P = log((k^(1/2))(R^(3/2)))
b) To rewrite the formula in exponential form, we need to convert the logarithmic equation back to exponential form:
Given the equation: log P = 1/2 (log k + 3 log R)
Exponential form states that if log(base a) x = y, then a^y = x.
Therefore, by applying this conversion:
P = 10^(1/2 log k + 3/2 log R)
Expanding the exponents:
P = 10^(log(k^(1/2)) + log(R^(3/2)))
Using the property: a^(b+c) = a^b * a^c
P = 10^(log(k^(1/2))) * 10^(log(R^(3/2)))
Simplifying further:
P = (k^(1/2)) * (R^(3/2))
Hence, the formula in exponential form is:
P = (k^(1/2)) * (R^(3/2))
c) To rewrite the formula solving for P, we will isolate P:
Given the equation: log P = 1/2 (log k + 3 log R)
First, we raise both sides of the equation to the power of the base 10:
10^(log P) = 10^(1/2 (log k + 3 log R))
Simplifying:
P = 10^(1/2 (log k + 3 log R))
Using the property: 10^(a * b) = (10^a)^b
P = (10^(log k + 3 log R))^(1/2)
Now, using the properties of logarithms and exponents:
P = (k * R^3)^(1/2)
Simplifying further:
P = (k * R^3)^(1/2)
Therefore, the formula solved for P is:
P = √(k * R^3)
a) To rewrite the formula as a single logarithm, we can use the properties of logarithms. First, recall that adding two logarithms with the same base is equivalent to multiplying their arguments. Additionally, multiplying a logarithm by a constant is representative of raising the argument to a power. Therefore, we can rewrite the given equation as:
log P = 1/2 (log k + 3 log R)
Using the properties mentioned earlier, we can simplify the right side of the equation as follows:
log P = 1/2 log (k) + 1/2 log (R^3)
Next, using the power rule of logarithms, which states that log (a^b) = b log (a), we can simplify further:
log P = log (k^(1/2)) + log (R^(3/2))
Finally, combining the two logarithms into a single logarithm, we have:
log P = log [(k^(1/2))(R^(3/2))]
Thus, the formula rewritten as a single logarithm is:
log P = log [(k^(1/2))(R^(3/2))]
b) To rewrite the formula in exponential form, we can use the definition of logarithms. The logarithmic equation log P = 1/2 (log k + 3 log R) can be transformed into exponential form as follows:
P = 10^(1/2(log k + 3 log R))
By applying the power rule of logarithms, which is the inverse of the logarithmic form, we raise the base (10) to the power of the entire right side of the equation. Therefore, the formula rewritten in exponential form is:
P = 10^(1/2(log k + 3 log R))
c) To rewrite the formula and solve for P, we can proceed as follows:
Start with the given equation:
log P = 1/2 (log k + 3 log R)
To isolate P, we first need to eliminate the logarithms. We can do this by raising both sides of the equation to the base 10:
10^(log P) = 10^(1/2 (log k + 3 log R))
Applying the power rule of logarithms to the left side:
P = 10^(1/2 (log k + 3 log R))
Since the expression inside the parentheses on the right side is a sum, we can use the power rule of logarithms to rewrite it:
P = 10^((1/2)log k + (3/2)log R)
Again applying the power rule of logarithms to each term:
P = 10^((log k)^(1/2)) * 10^((log R)^3/2)
Now, we can simplify further:
P = √k * (R^3/2)
Therefore, the formula rewritten to solve for P is:
P = √k * (R^3/2)