Question

Write the following expression as a sum and/or difference of logarithms. Express powers as factors.

logᵦ(u⁵v⁸) u > 0, v > 0

Answer 1

The expression logᵒ(u⁵v⁸) can be written as a sum of logarithms using the properties of logarithms: logᵒ(u⁵v⁸) = 5 ⋅ logᵒ(u) + 8 ⋅ logᵒ(v).

Step-by-step explanation:

You want to write the expression logᵒ(u⁵v⁸) as a sum and/or difference of logarithms. To do this, you can use the properties of logarithms:

1. The logarithm of a product equals the sum of the logarithms of the individual factors: log(xy) = log(x) + log(y).
2. The logarithm of a quotient equals the difference between the logarithms of the numerator and the denominator: log(x/y) = log(x) - log(y).
3. The logarithm of a power equals the exponent times the logarithm of the base: log(x^n) = n ⋅ log(x).

Applying these properties to the given expression, we have:

logᵒ(u⁵v⁸) = logᵒ(u⁵) + logᵒ(v⁸)
= 5 ⋅ logᵒ(u) + 8 ⋅ logᵒ(v)

So the expression is written as a sum of logarithms, with the powers becoming factors in front of the logarithms.