If log10 a= x and log10 b=y, express log10 [100a^(3) b^(-1/2) ÷b² in terms of x and y
2+3x-5/2y
We need to first clarify whether 5/2 is being subtracted from y or is a part of the denominator. Assuming that the expression is:
2 + 3x - (5/2)y
We can express it as:
2 + 3x - 2.5y
Therefore, the expression 2 + 3x - 5/2y can be simplified to 2 + 3x - 2.5y.
2 + 3x - 5/2 y
The expression 2 + 3x - 5/2 y cannot be simplified any further without additional information or context. It is already in its simplest form.
To express log10 [100a^(3) b^(-1/2) ÷b²] in terms of x and y, we can use logarithmic identities and properties of exponents. Let's break down the expression step by step:
1. Start with the given expression: log10 [100a^(3) b^(-1/2) ÷b²].
2. Apply the rules of logarithms: First, let's simplify the numerator and denominator separately. In the numerator, 100 and a^3 can be combined using exponent properties as 100 * a^3 = 100a^3. In the denominator, b^(-1/2) ÷ b^2 can be written using the quotient property of logarithms as b^(-1/2 - 2) = b^(-5/2).
Now, our expression becomes: log10 [(100a^3 * b^(-5/2))].
3. Next, let's use the power rule of logarithms: log b^n = n * log b.
Applying the power rule to the expression, we get: (log10 100a^3) + (log10 b^(-5/2)).
4. Simplify each term further:
- For the first term, log10 100a^3, we can apply the properties of logarithms. Using the product rule, log10 (ab) = log10 a + log10 b. We have log10 (100a^3) = log10 100 + log10 a^3. Since log10 100 equals 2 (as 10² = 100), we can simplify further to get: 2 + 3(log10 a) = 2 + 3x.
- For the second term, log10 b^(-5/2), we can use the power rule and bring the exponent down: (-5/2) * (log10 b) = (-5/2) * y = -5y/2.
5. Combine the simplified terms: (2 + 3x) - (5y/2).
Therefore, log10 [100a^(3) b^(-1/2) ÷b²] can be expressed as (2 + 3x) - (5y/2).
Using the rules of logarithms, we can simplify the expression as follows:
log10 [100a^(3) b^(-1/2) ÷b²]
= log10 [100a^(3) b^(-1/2)] - log10 b²
= log10 100 + log10 a^(3) - (log10 b^(1/2))^2 - 2 log10 b
= 2 + 3x - y - 2y
= 2 + 3x - 3y
Therefore, log10 [100a^(3) b^(-1/2) ÷b²] can be expressed in terms of x and y as 2 + 3x - 3y.