Express 2 lg 3√x - lg 4/3x^2 + log(100)4x^3 in the form a lg x + lg b ,where a and b are constants .
assuming log(100) means log base 100,
and assuming 4/3x^2 means 4/(3x^2),
2 lg 3√x - lg 4/3x^2 + log(100)4x^3
lg 9x - lg 4/3x^2 + 1/2 lg 4x^3
lg 9 + lg x - lg 4 + lg 3 + 2lg x + lg 2 + 3/2 lg x
(1+2+3/2)lg x + lg(9/4*3*2)
9/2 lg x + lg 27/2
To express 2 lg(3√x) - lg(4/3x^2) + log(100)4x^3 in the form a lg(x) + lg(b), we need to apply logarithmic properties and simplify the expression.
First, let's simplify each term individually:
1. Simplify lg(3√x):
Recall that the square root (√) can be rewritten as a fractional exponent. So, 3√x can be written as x^(1/3). Therefore, lg(3√x) becomes lg(x^(1/3)). By applying the logarithmic property, we can bring the exponent out front, resulting in (1/3) lg(x).
2. Simplify lg(4/3x^2):
We can rewrite this term as a subtraction using logarithmic properties. lg(4/3x^2) is equivalent to lg(4) - lg(3x^2). Again, applying the property, we have lg(4) - 2 lg(x) - lg(3).
3. Simplify log(100)4x^3:
Combining logarithmic properties, we can rewrite this as 3 log(4x). Since log(100) = 2 (as 100 = 10^2), we multiply the exponent by 3.
Now, let's substitute these simplifications back into the original expression:
2 lg(3√x) - lg(4/3x^2) + log(100)4x^3
= 2 (1/3) lg(x) - (lg(4) - 2 lg(x) - lg(3)) + 3 log(4x)
= (2/3) lg(x) - lg(4) + 2 lg(x) + lg(3) + 3 log(4x)
= (2/3) lg(x) + 2 lg(x) - lg(4) + lg(3) + 3 log(4x)
= (8/3) lg(x) - lg(4) + lg(3) + 3 log(4x)
Now, we can rewrite the expression in the form a lg(x) + lg(b):
= (8/3) lg(x) + (lg(3) - lg(4)) + 3 log(4x)
Therefore, the original expression can be expressed as (8/3) lg(x) + lg(3/4) + 3 log(4x).