How do you evaluate logarithms?

The basic question behind every logarithm

When evaluating logarithms, the key is knowing that a logarithm is simply an exponent in disguise. The expression log_b(x) = y means exactly the same thing as b^y = x. The small lower number, b, is the base. The number inside, x, is the argument. The output, y, is the power you need.

To evaluate log_2(8), say out loud: 2 raised to what power equals 8? Since 2 * 2 * 2 = 8, that power is 3. Therefore, log_2(8) = 3. Thinking of the log sign as a question about exponents makes solving basic problems quick and straightforward.

Steps to evaluate logarithms without a calculator

First, set the logarithmic expression equal to a variable, such as y. For instance, write log_3(81) = y.

Second, rewrite the equation in exponential form by moving the base under the other side: 3^y = 81.

Third, write both sides using the same base. Since 81 equals 3^4, your equation becomes 3^y = 3^4. Because the bases match, set the exponents equal: y = 4. This method works cleanly for whole powers, fractions, and negative powers like log_5(1/25) = -2.

Allowed values and calculator keys

A logarithm has strict rules: the base must be positive and not equal to 1, and the number inside must always be greater than zero. You cannot take the logarithm of zero or a negative number in real numbers, because no positive base raised to a real power produces zero or a negative result.

When no base is written, such as log(100), the base is an invisible 10. This is the common log, found on calculators as the LOG key. The natural logarithm, written ln(x), uses the constant e (about 2.718) as its base and has its own LN key.