Express x in terms of a,b and c.

log x = 1/2 (log a + log b - log c)
Please solve and explain how to do this type of problem, thank you!

duplicate

To solve

log x = 1/2 (log a + log b - log c)
we need to know some properties of logarithms, namely:

1. log(a)+log(b) = log(ab)
2. (1/2)log(a)=log(a-1/2)=log(√(a))
3. eln x = x, or
10log10 x = x

Proceeding to simplify the right-hand-side,
log x = 1/2 (log a + log b - log c)
= (1/2)log a + (1/2)log b + (1/2)log c
= log(√a) + log(√b) + log(√c)
= log(√a √b √c)
= log(√(abc))

Assuming log() stands for logarithm to the base e,
elog x = elog(√(abc))
x = √(abc)

2nd rule of logarithm should read:

2. (1/2)log(a)=log(a1/2)=log(√(a))

and the solution has to be corrected because of an erroneous sign:

Proceeding to simplify the right-hand-side,
log x = 1/2 (log a + log b - log c)
= (1/2)log a + (1/2)log b - (1/2)log c
= log(√a) + log(√b) - log(√c)
= log(√a √b / √c)
= log(√(ab/c))

Assuming log() stands for logarithm to the base e,
elog x = elog(√(ab/c))
x = √(ab/c)

To express x in terms of a, b, and c, we'll first use the properties of logarithms to simplify the equation. Then we'll isolate x.

Let's begin by applying the logarithmic property:

log x = 1/2 (log a + log b - log c)

Now, we can use the property that states the sum of logarithms is equal to the logarithm of the product:

log x = 1/2 (log (a * b / c))

Next, we can use another logarithmic property that states the logarithm of a division is the difference of the logarithms:

log x = 1/2 (log (a * b) - log c)

Furthermore, we can distribute the 1/2 exponent inside the parentheses:

log x = 1/2 log (a * b) - 1/2 log c

Now, since log x is equal to the expression on the right-hand side, we can write:

x = 10^(1/2 log (a * b) - 1/2 log c)

Finally, we can simplify further:

x = 10^(log ((a * b)^(1/2) / c^(1/2)))

Using the properties of exponential and logarithmic functions, we can simplify:

x = ((a * b)^(1/2)) / (c^(1/2))

Simplifying the square roots:

x = (sqrt(a * b)) / (sqrt(c))

Thus, x is expressed in terms of a, b, and c as:

x = (sqrt(a * b)) / (sqrt(c))

That's the answer to the equation. Following the steps, we used logarithmic properties to simplify and then isolate x in terms of a, b, and c.