Question

Use the product, quotient, and power rules of logarithms to rewrite the expression as a single logarithm. Assume that all variables represent positive real numbers.
log base two of eight minus log base two of z.



log base two of z divided by eight.


log base two of eight divided by z.


log base two of eight minus z.


log base four of eight divided by z.

Answer 1

Answer:

2log₂(x) +3log₂(y) -log₂(9)

Step-by-step explanation:

The applicable rules of logarithms are ...

... log(ab) = log(a) + log(b)

... log(a^b) = b·log(a)

... log(a/b) = log(a) -log(b)

Note that these are essentially all the same rule, expressed in different forms. For example, ...

... log(a·a) = log(a²) = log(a) +log(a) = 2·log(a)

The same sort of result obtains with larger exponents or exponents that are negative.

... log(a/b) = log(a·b⁻¹) = log(a) +(-1)log(b) = log(a) -log(b)

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Applying these to your problem, you have ...

... log₂(x²y³/9) = log₂(x²) +log₂(y³) -log₂(9)

... = 2log₂(x) +3log₂(y) -log₂(9)