log r - log r^3 - log r^5 =8√2 , if all the logarithms are in base 2, find r.
Using the properties of logarithms, we can simplify the equation as follows:
log r - log r^3 - log r^5 = 8√2
log r - 3log r - 5log r = 8√2
-7log r = 8√2
log r = -8√2 / 7
Since the logarithms are in base 2, we can rewrite the equation as:
r = 2^(-8√2 / 7)
Therefore, r ≈ 0.7027.
To solve the equation, let's simplify the expression first using the properties of logarithms.
log r - log r^3 - log r^5 = 8√2
Using the properties of logarithms, we can rewrite the equation as:
log2(r) - log2(r^3) - log2(r^5) = 8√2
Next, we can use the following properties of logarithms to simplify further:
1. log(a) - log(b) = log(a/b)
2. log(a^b) = b * log(a)
Applying these properties, we can rewrite the equation as:
log2(r) - 3 * log2(r) - 5 * log2(r) = 8√2
Simplifying the expression further:
-3 * log2(r) - 5 * log2(r) = 8√2 - log2(r)
Combining the terms:
-8 * log2(r) = 8√2 - log2(r)
Now, let's isolate the term containing the logarithm:
-8 * log2(r) + log2(r) = 8√2
Simplifying:
-7 * log2(r) = 8√2
To solve for r, we need to eliminate the logarithm. We know that log2(r) is the exponent we need to raise 2 to in order to get r. Therefore, we can rewrite the equation as follows:
2^(-7 * log2(r)) = 2^(8√2)
Simplifying the right side:
2^(-7 * log2(r)) = 2^(8√2)
2^(-7 * log2(r)) = 2^(√128)
2^(-7 * log2(r)) = 2^(√(64 * 2))
2^(-7 * log2(r)) = 2^((√64) * (√2))
2^(-7 * log2(r)) = 2^(8 * √2)
Since the bases are the same, we can set the exponents equal to each other:
-7 * log2(r) = 8 * √2
Dividing both sides by -7:
log2(r) = -(8/7) * √2
Exponentiating both sides with base 2:
2^(log2(r)) = 2^(-(8/7) * √2)
Simplifying the left side:
r = 2^(-(8/7) * √2)
Thus, r is approximately equal to 0.7344.
To solve the equation, we'll start by using the logarithmic properties to simplify the equation. The properties we'll use are:
1. log(x^a) = a * log(x) (logarithm of a power)
2. log(a) - log(b) = log(a / b) (difference of logarithms)
Applying these properties to the equation, we have:
log(r) - log(r^3) - log(r^5) = 8√2
Using property 1, we can rewrite log(r^3) and log(r^5) as:
3 * log(r) and 5 * log(r), respectively. Substituting these values into the equation, we get:
log(r) - 3 * log(r) - 5 * log(r) = 8√2
Combining like terms, we have:
-7 * log(r) = 8√2
Now, we can isolate log(r) by dividing both sides of the equation by -7:
log(r) = (8√2) / -7
Next, we need to convert this logarithmic equation into an exponential equation. In base 2 logarithms, the base is 2. So, we can rewrite the equation as:
r = 2^[(8√2) / -7]
Simplifying the right side of the equation, we have:
r ≈ 2^(-8√2 / 7)
Therefore, the approximate value of r is 2 raised to the power of -8√2 divided by 7.