Evaluate logarithms with integer values

A logarithm is evaluated as the exponent to which a valid base must be raised to produce the given positive argument: for example, $\log_2 8=3$ , $\log_{10}1000=3$ , and $\log_2(1/8)=-3$ . The understanding includes $\log_b1=0$ , bases $b>0$ with $b\ne1$ , and identifying exact integer powers rather than treating a logarithm as multiplication; decimal approximations, change-of-base methods, and more general noninteger evaluations are beyond this scope.

Detailed Explanation: 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.

Learn by doing: Evaluate logarithms with integer values

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Logarithms - Solve Exponent Equation (Integers)

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