If log10 a=x and log10 b=y, express log10 [ 100a^(3)b^(-2/1) ÷ b^(2) ] in terms of x and y
We can write the given expression as:
2 + 3x - (5/2)y
Or,
2 + 3x - 2.5y
Thus, the expression 2+3x - (5/2 y) simplified is 2 + 3x - 2.5y.
2+3x - (5/2 y)
In order to express the logarithm expression log10 [ 100a^(3)b^(-2/1) ÷ b^(2) ] in terms of x and y, let's simplify the given expression step by step:
Step 1: Distribute the power of a and simplify
100a^(3)b^(-2/1) = 100(a^3)(b^(-2))
= 100a^3 b^(-2)
Step 2: Apply the logarithmic properties
log10 [ 100a^(3)b^(-2/1) ÷ b^(2) ] = log10(100a^3 b^(-2)) - log10(b^2)
Step 3: Apply the power law of logarithm
log10 [ 100a^(3)b^(-2/1) ÷ b^(2) ] = log10(100) + log10(a^3) + log10(b^(-2)) - log10(b^2)
Step 4: Simplify logarithmic terms
Since log10 a = x and log10 b = y, we have:
log10 [ 100a^(3)b^(-2/1) ÷ b^(2) ] = log10(100) + 3log10(a) - 2log10(b) - 2log10(b)
Step 5: Substitute the values of x and y
Since log10 a = x and log10 b = y, we replace them in the expression:
log10 [ 100a^(3)b^(-2/1) ÷ b^(2) ] = log10(100) + 3x - 2y - 2y
Step 6: Simplify the expression
log10 [ 100a^(3)b^(-2/1) ÷ b^(2) ] = log10(100) + 3x - 4y
Therefore, log10 [ 100a^(3)b^(-2/1) ÷ b^(2) ] can be expressed in terms of x and y as log10(100) + 3x - 4y.
Using the laws of logarithms:
log10 [ 100a^(3)b^(-2/1) ÷ b^(2) ]
= log10 [100a^(3) b^(-2/1) ] - log10 [b^(2)]
= log10 [100] + log10 [a^(3)] - 2 log10 [b] - log10 [b^(2)]
= 2 + 3x - 2y - 2
= 3x - 2y