Question

Express the following as a single logarithm and simplify: 5log₂x-(1)/(2)log₂(3x-4)-3log₂(5x+1)

Answer 1

To simplify 5log₂x - (1)/(2)log₂(3x-4) - 3log₂(5x+1) as a single logarithm, apply the logarithm properties of products, quotients and exponents to rewrite the expression. The result is log₂(x⁵/((3x-4)¹⁄₂(5x+1)³)).

Step-by-step explanation:

To express the equation 5log₂x - (1)/(2)log₂(3x-4) - 3log₂(5x+1) as a single logarithm and simplify, we can use the logarithm properties:

1. The logarithm of a product of two numbers is the sum of the logarithms of the two numbers: logb(xy) = logb(x) + logb(y).
2. The logarithm of a quotient is the difference between the logarithms of the numerator and the denominator: logb(x/y) = logb(x) - logb(y).
3. The logarithm of a number raised to an exponent is the product of the exponent and the logarithm of the number: logb(xn) = n · logb(x).

Using these properties, we can rewrite the equation as:

log₂(x5) - log₂((3x-4)1/2) - log₂((5x+1)3)

Next, we combine the logarithms into one:

log₂(x5/((3x-4)1/2(5x+1)3)

Now, this is our simplified single logarithm.