Express as a single logarithm.
1/2 log base b of (x-2)- log base b of y + (3 log base b of z) = log base 6 of (x-2)^1/2
(1/2) logb(x-2) - logby + 3logbz = log6(x-2)^(1/2)
logb [ (x-2)^(1/2) z/y] = log6(x-2)^(1/2)
logb ( (z/y)√(x-2) ) = log6 √(x-2)
are you sure the right side was not supposed to be
logb ?
oops, didn't notice the 3 coefficient until I checked the display of this post.
should have been: z^3 in the last two lines.
logb (x^3/y)logb(x-2) = log6 √(x-2)
To express the given expression as a single logarithm, we will use logarithmic properties, including the product rule and the power rule.
First, let's simplify each term individually:
1/2 log base b of (x-2):
Using the power rule, we can write this as log base b of (x-2)^(1/2).
-log base b of y:
This term remains unchanged.
(3 log base b of z):
Using the power rule, we can rewrite this as log base b of z^3.
Now, let's combine these terms:
1/2 log base b of (x-2) - log base b of y + (3 log base b of z) =
log base b of (x-2)^(1/2) - log base b of y + log base b of z^3
Since all the terms are in the same base (b), we can simplify further using the product rule:
log base b of (x-2)^(1/2) - log base b of y + log base b of z^3 =
log base b of [(x-2)^(1/2) * z^3 / y]
Finally, let's convert the expression to a different base, log base 6:
log base b of [(x-2)^(1/2) * z^3 / y] =
(log base b of [(x-2)^(1/2) * z^3 / y]) / (log base b of 6)
Therefore, the expression as a single logarithm in log base 6 is:
(log base b of [(x-2)^(1/2) * z^3 / y]) / (log base b of 6)
To express the given equation as a single logarithm, we can start by using the properties of logarithms to combine the terms involving the same base.
First, let's combine the terms involving log base b:
1/2 log base b of (x-2) - log base b of y + 3 log base b of z
By applying the properties of logarithms, we can rewrite this expression as a single logarithm:
log base b of [(x-2)^(1/2)] / (y) * (z^3)
Next, let's express the right-hand side of the equation as a logarithm:
log base 6 of (x-2)^(1/2)
Using the property of logarithms, we can convert this equation to exponential form:
(x-2)^(1/2) = 6
Now we can square both sides of the equation to eliminate the square root:
[(x-2)^(1/2)]^2 = 6^2
x - 2 = 36
x = 38
Therefore, the solution to the original equation is x = 38.