Skill: Expand logarithmic expressions

To expand a logarithmic expression, rewrite it using these rules:

$$\log_b(xy)=\log_b x+\log_b y$$ $$\log_b\left(\frac{x}{y}\right)=\log_b x-\log_b y$$ $$\log_b(x^r)=r\log_b x$$

These rules apply to real logarithms when $b>0$, $b\ne 1$, and all logarithm arguments are positive.

Example

Expand:

$$\log_2\left(\frac{8x^3\sqrt{y}}{z^2}\right)$$

Assume $x>0$, $y>0$, and $z>0$.

Step 1: Separate the quotient.

Use the quotient rule:

$$\log_2\left(\frac{8x^3\sqrt{y}}{z^2}\right) = \log_2(8x^3\sqrt{y})-\log_2(z^2)$$

Step 2: Expand the product in the numerator.

Use the product rule:

$$\log_2(8x^3\sqrt{y}) = \log_2 8+\log_2(x^3)+\log_2(\sqrt{y})$$

So the expression becomes

$$\log_2 8+\log_2(x^3)+\log_2(\sqrt{y})-\log_2(z^2)$$

Step 3: Move powers in front as coefficients.

Since $\sqrt{y}=y^{1/2}$,

$$\log_2(x^3)=3\log_2 x$$ $$\log_2(\sqrt{y})=\log_2(y^{1/2})=\frac12\log_2 y$$

and

$$\log_2(z^2)=2\log_2 z$$

Also, $\log_2 8=3$ because $2^3=8$.

Therefore, the expanded expression is

$$\boxed{3+3\log_2 x+\frac12\log_2 y-2\log_2 z}$$

Remember that logarithms can be expanded over products and quotients, but not over sums or differences. For example,

$$\log_b(x+y)\ne \log_b x+\log_b y$$