Skill: Condense logarithmic expressions

To condense logarithms, reverse the logarithm properties:

• $k\log_b M=\log_b(M^k)$
• $\log_b M+\log_b N=\log_b(MN)$
• $\log_b M-\log_b N=\log_b\left(\frac{M}{N}\right)$

The logarithms must have the same base, and every argument must be positive. Also, a sum of logarithms becomes the logarithm of a product, not a sum.

Example

Condense the expression:

$$2\log_3 x+\log_3(x-1)-\log_3 5$$

Assume $x>1$, so both $x$ and $x-1$ are positive.

Step 1: Move the coefficient into the first logarithm as an exponent.

$$2\log_3 x=\log_3(x^2)$$

Now the expression is

$$\log_3(x^2)+\log_3(x-1)-\log_3 5$$

Step 2: Combine the sum using the product property.

$$\log_3(x^2)+\log_3(x-1)=\log_3\bigl(x^2(x-1)\bigr)$$

So we have

$$\log_3\bigl(x^2(x-1)\bigr)-\log_3 5$$

Step 3: Combine the difference using the quotient property.

$$\boxed{\log_3\left(\frac{x^2(x-1)}{5}\right)}$$

Thus,

$$2\log_3 x+\log_3(x-1)-\log_3 5 =\log_3\left(\frac{x^2(x-1)}{5}\right).$$