Suppose that log2 = a and log3 = b
Solve for x in terms of a and b:
6^x = 10/3 - 6^-x
Parentheses are not needed for fractions in type-set texts because it is graphically clear. When transferred to a one-line expression, parentheses are required around the numerator and denominator.
I assume the question is as follows, which is different from what you posted because of the missing parentheses.
6^x=10/(3-6^(-x)) ....(1)
From log(2)=a, and log(3)=b,
we deduce that
ea=2, and
eb=3.
cross multiply (1), assume 3≠6^(-x)
6^x(3-6^(-x))=10
3*6^x - 6^0 = 10 .... (6^0 = 1)
3*6^x = 11
6^x = 11/3
take log on both sides
x(log(2)+log(3) = log(11)-log(3)
x = (log(11)-b)/(a+b)
To solve for x in terms of a and b, let's first rewrite the equation using the properties of logarithms.
Given:
log₂ = a
log₃ = b
We know that logarithmic identities state:
1. logₐ(b) = log(b) / log(a)
2. log(a * b) = log(a) + log(b)
3. log (a / b) = log(a) - log(b)
Now, let's rewrite the equation using these identities:
6^x = 10/3 - 6^-x
To simplify the equation, we'll express the constants on the right side of the equation using the given logarithmic identities.
First, express 6^-x in terms of logarithms:
6^-x = 1 / 6^x
Now, rewrite both sides of the equation using logarithms to the base 2 and the base 3:
2^(log₂(6^x)) = 2^(log₂(10/3 - 1 / 6^x))
3^(log₃(6^x)) = 3^(log₃(10/3 - 1 / 6^x))
Using the properties of logarithms from step 1, we can rewrite the equation:
2^(x * log₂(6)) = 2^(log₂(10/3) - log₂(6^x))
3^(x * log₃(6)) = 3^(log₃(10/3) - log₃(6^x))
Using the property from step 2, we have:
2^x * 2^(log₂(6)) = 2^(log₂(10/3)) * 2^(-x * log₂(6))
3^x * 3^(log₃(6)) = 3^(log₃(10/3)) * 3^(-x * log₃(6))
Now, let's replace log₂(6) and log₃(6) using the given values a and b:
2^x * 2^a = 2^(log₂(10/3)) * 2^(-x * a)
3^x * 3^b = 3^(log₃(10/3)) * 3^(-x * b)
Using the property from step 2 again, we can combine the exponents:
2^(x + a) = 2^(log₂(10/3) - x * a)
3^(x + b) = 3^(log₃(10/3) - x * b)
Finally, after equating the exponents, we can solve for x:
x + a = log₂(10/3) - x * a
x + b = log₃(10/3) - x * b
Rearranging the equations:
x * (1 + a) = log₂(10/3) - a
x * (1 + b) = log₃(10/3) - b
Finally, solve for x by dividing both sides by (1 + a) and (1 + b):
x = (log₂(10/3) - a) / (1 + a)
x = (log₃(10/3) - b) / (1 + b)
So, x in terms of a and b is:
x = (log₂(10/3) - a) / (1 + a)
x = (log₃(10/3) - b) / (1 + b)