Given log(base a)27=b. Find log(base sqrt 3)(a^1/6)
a^b = 3^3
3^(1/2)^x = a^(1/6)
3^3^x = a
3^3 = a^(1/x)
x = 1/b
Not sure I get this step, since ^ is right-associative:
3^(1/2)^x = a^(1/6)
3^3^x = a
As written, that means 3^(3^x)
I was thinking
(3^(1/2))^x = a^(1/6)
3^(x/2) = a^(1/6)
3^(3x) = a
3^3 = a^(1/x) = a^b
1/x = b
x = 1/b
To find log(base √3)(a^(1/6)), we can use the change of base formula. The change of base formula states that log(base a)(b) can be written as log(base c)(b) divided by log(base c)(a), where c can be any positive number other than 1.
Step 1: Find log(base c)(a):
Since we are given the equation log(base a)(27) = b, we can rewrite it as:
a^b = 27
To find log(base c)(a), we need to convert the given equation to base c. Taking the logarithm of both sides, we get:
log(base c)(a^b) = log(base c)(27)
Now, using the logarithmic rule, we can bring the exponent down to the front:
b * log(base c)(a) = log(base c)(27)
Dividing both sides by b, we get:
log(base c)(a) = log(base c)(27) / b
Step 2: Find log(base c)(27):
To find log(base c)(27), we can use the change of base formula again. This time, we'll use c = √3 as the base, as given in the problem.
log(base √3)(27) = log(base √3)(3^3)
Using the logarithmic rule, we can bring the exponent down to the front:
3 * log(base √3)(3) = 3 * 1
log(base √3)(27) = 3
Step 3: Substitute the values into the equation:
Now that we know log(base c)(a) and log(base c)(27), we can substitute them into the equation log(base √3)(a^(1/6)):
log(base √3)(a^(1/6)) = log(base √3)(27) / b
Substituting the values we found in step 2, we have:
log(base √3)(a^(1/6)) = 3 / b
Therefore, log(base √3)(a^(1/6)) is equal to 3 / b.