Skill: Evaluate logarithms with integer values

A logarithm asks:

What exponent should the base be raised to in order to get the argument?

In symbols,

$$\log_b a=x \quad \text{means} \quad b^x=a.$$

The base must satisfy $b>0$ and $b\ne 1$, and the argument $a$ must be positive.

Example

Evaluate:

$$\log_3\left(\frac{1}{27}\right)$$

Step 1: Rewrite the question using the definition of a logarithm.

Let

$$\log_3\left(\frac{1}{27}\right)=x.$$

This means

$$3^x=\frac{1}{27}.$$

Step 2: Express the argument as a power of the base.

Since

$$27=3^3,$$

its reciprocal is

$$\frac{1}{27}=\frac{1}{3^3}=3^{-3}.$$

Therefore,

$$3^x=3^{-3}.$$

Step 3: Compare the exponents.

Because the bases are the same,

$$x=-3.$$

So,

$$\boxed{\log_3\left(\frac{1}{27}\right)=-3}.$$

The answer is negative because a negative exponent produces a fraction. For example, $\log_b 1=0$ because $b^0=1$. Also remember that a logarithm is asking for an exponent, not multiplying the base by the argument.