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.