Introduction
A logarithm answers an exponent question: what power of a specified base produces this number? When the argument is already an exact integer power of the base, the logarithm can be evaluated without a calculator.
For example, because \(2^3=8\), the exponent that makes \(2\) equal \(8\) is \(3\). Because \(10^3=1000\), the exponent that makes \(10\) equal \(1000\) is also \(3\). The key is to identify the base and the exponent precisely.
Definition and Conditions
The base \(b\) must be positive and different from \(1\). The argument \(x\), which is the number inside the logarithm, must be positive. These conditions ensure that the logarithm is defined as the inverse of the exponential function with base \(b\). Since that exponential function is one-to-one for an allowed base, there is only one exponent \(y\) that gives the argument \(x\).
In particular, a logarithm is not an instruction to divide two numbers. Its value is an exponent. For instance, \(\log_2(8)\) asks which exponent on \(2\) produces \(8\), not what \(8\) divided by \(2\) equals.
Recognizing Exact Values
Suppose the argument can be written as \(b^n\), where \(n\) is an integer. The defining equivalence gives an immediate result:
This works for positive, zero, and negative integers \(n\), provided the base satisfies \(b>0\) and \(b\ne 1\). Integer exponents are interpreted using the exponent conventions established earlier: \(b^0=1\), and for \(n>0\), \(b^{-n}=1/b^n\).
Why the Exponent-Matching Method Works
The method follows directly from the definition, rather than from a separate calculation rule. If \(x=b^n\), then \(n\) is an exponent that produces \(x\). The logarithm \(\log_b(x)\) is defined to be the exponent that produces \(x\). Because \(x\) has exactly one such exponent for an allowed base, that exponent must be \(n\).
In \(\log_b(x)\), the base is \(b\), the number being raised to a power.
Find an integer \(n\) such that \(b^n=x\), using the exponent definitions when the exponent is zero or negative.
Write \(\log_b(x)=n\), then verify that \(b^n\) really equals \(x\).
This reasoning applies whether \(b\) is greater than \(1\) or lies between \(0\) and \(1\). When \(0<b<1\), increasing the exponent makes \(b^x\) smaller rather than larger, but the exponential function is still one-to-one. The exponent is therefore still unique.
Worked Examples
Worked Example: Evaluating \(\log_2(1/8)\)
We want an exponent \(n\) such that \(2^n=1/8\). First write the fraction as a power of \(2\):
Thus the argument is \(2^{-3}\), so the exponent that produces it from base \(2\) is \(-3\):
Check: Convert the proposed logarithm back to exponential form:
The result matches the original argument, so the exact value is \(-3\).
Worked Example: Evaluating \(\log_{10}(1000)\)
We want an exponent \(n\) such that \(10^n=1000\). Write \(1000\) as a power of \(10\):
Therefore, the exponent is \(3\):
Check: The defining exponential equation is satisfied:
So the exact value is \(3\). A logarithm with base \(10\) is called a common logarithm. The base is sometimes omitted in common-log notation, but writing \(\log_{10}\) makes the base explicit.
Worked Example: Evaluating \(\ln(e^3)\)
The notation \(\ln(x)\) means \(\log_e(x)\), the logarithm with base \(e\). Here the argument is already a third power of that base:
Hence the exponent that produces \(e^3\) from base \(e\) is \(3\):
Check: The exponential form of the answer gives back the argument:
Thus \(\ln(e^3)=3\). The notation \(\ln\) is a special case of the same logarithm definition, with base \(e\).
Worked Example: Evaluating \(\log_{1/3}(27)\)
This example has a base between \(0\) and \(1\), so a negative exponent is needed to produce an argument greater than \(1\). We seek \(n\) such that \((1/3)^n=27\). Since \(27=3^3\), use the negative-exponent definition:
Therefore, \(-3\) is the exponent that produces \(27\):
Check: Substituting the answer into the defining equation yields \((1/3)^{-3}=27\), as required.
Zero and Negative Exponents
Two common cases follow from the exponent definitions. For every allowed base \(b\), \(b^0=1\), so the exponent that produces \(1\) is \(0\). Also, if \(n\) is a positive integer, \(b^{-n}=1/b^n\), so reciprocals of positive integer powers correspond to negative logarithm values.
These statements require \(b>0\) and \(b\ne1\); the argument \(1\) or \(1/b^n\) is positive under those conditions. For example, \(1/16=2^{-4}\), so \(\log_2(1/16)=-4\). The sign is determined by the exponent that actually produces the argument, not by a guess based only on how the number looks.
Common Logarithm Notation and Scope
The base should always be identified before evaluating. In common-log notation, \(\log_{10}(x)\) is often written simply as \(\log(x)\). In natural-log notation, \(\log_e(x)\) is written as \(\ln(x)\). When a base is displayed explicitly, the definition is unambiguous:
| Expression | Base | Exponent question |
|---|---|---|
| \(\log_2(8)\) | \(2\) | What power of \(2\) is \(8\)? |
| \(\log_{10}(1000)\) | \(10\) | What power of \(10\) is \(1000\)? |
| \(\ln(e^3)\) | \(e\) | What power of \(e\) is \(e^3\)? |
For exact evaluation in this lesson, the argument is recognizable as an integer power of the same base. If it is not, exponent matching may not produce an integer answer; that does not mean the logarithm is undefined. It means only that this exact integer-power method does not directly identify its value.
A Common Mistake
A frequent error is to treat a negative exponent as if it meant a negative number. For example, one might incorrectly reason that \(2^{-3}=-8\), and then claim that \(\log_2(1/8)\) is not \(-3\).
The correction is to apply the definition of a negative exponent: take the reciprocal of the positive power. The minus sign in the exponent does not make the resulting value negative.
Therefore \(2^{-3}=1/8\), and the correct logarithm value is \(\log_2(1/8)=-3\). Checking by converting back to exponential form catches the sign error immediately.
Why Exact Logarithm Values Matter
Exact values preserve the relationship between exponential and logarithmic expressions without introducing rounding. That is useful when checking calculations, describing a quantity in a different form, or preparing to solve equations involving exponents. At this stage, the essential skill is modest but important: identify the base, match the argument to an exact power, and verify the exponent.
Keep the definition in view. A logarithm is an exponent, its argument must be positive, and its base must be positive and unequal to \(1\). If the proposed answer satisfies the corresponding exponential equation, the exact value is confirmed.
Check Your Understanding
For each question, identify the exponent and verify the result by writing the corresponding exponential equation.
- Evaluate \(\log_3(81)\).
- Evaluate \(\log_5(1/25)\).
- Evaluate \(\log_{10}(1)\).
- Evaluate \(\ln(e^{-2})\).
- Evaluate \(\log_{1/2}(4)\).
- State why \(\log_2(-8)\) is not defined as a real logarithm.
