Why the Number \(e\) Matters
The number \(e\) is a particular positive real number, approximately \(2.71828\). It is important because it serves as the base of the natural exponential function, \(e^x\), and the logarithm that reverses that function is called the natural logarithm, written \(\ln(x)\).
You have already met logarithms as inverses of exponential functions and used logarithms with different bases. The natural logarithm is not a new kind of operation: it is the logarithm with the particular base \(e\). This choice of base gives the inverse pair \(e^x\) and \(\ln(x)\) a standard, compact notation used throughout mathematics and its applications.
Defining \(e\) and the Natural Logarithm
The number \(e\) is an irrational constant. Its decimal expansion begins \(2.718281828\ldots\), so its decimal representation continues without terminating or repeating. One standard characterization is
This limit is one way to describe the number; for calculations here, a sufficiently accurate decimal approximation is often useful. The base \(e\) is positive and is not equal to \(1\), so it is a valid logarithm base.
The domain restriction follows from the exponential equation in the definition: \(e^y\) is positive for every real \(y\), so it can equal only a positive \(x\). Conversely, for every positive \(x\), the logarithm base \(e\) is defined and produces a real exponent.
The Inverse Identities
Because the natural logarithm and the exponential function undo one another, their composition gives two important identities. The order of the operations matters: one identity begins with any real exponent, while the other begins with a positive input to the logarithm.
These statements are instances of the inverse-function rule. The exponential function \(y=e^x\) has domain \(\mathbb{R}\) and range \((0,\infty)\); its inverse therefore has domain \((0,\infty)\) and range \(\mathbb{R}\). The two identities express that each function returns the input to the other.
The identities can also be checked directly from the definition. Since \(e^x\) is positive, \(\ln(e^x)\) is defined. The definition says that \(\ln(e^x)\) is the exponent to which \(e\) must be raised to obtain \(e^x\); that exponent is \(x\). In the other direction, if \(x>0\), then \(y=\ln(x)\) is defined and satisfies \(e^y=x\). Substituting \(y=\ln(x)\) gives \(e^{\ln(x)}=x\).
Some exact values follow immediately. The exponent on \(e\) that produces \(e\) is \(1\), so \(\ln(e)=1\). The exponent that produces \(1\) is \(0\), since \(e^0=1\), so \(\ln(1)=0\). More generally, \(\ln(e^a)=a\) for every real \(a\).
Worked Example: Evaluate Expressions Involving \(e\) and \(\ln\)
Evaluate \(\ln(e^{-2})\), \(e^{\ln(5)}\), and \(\ln(1)\), and check each result.
First, the input to the logarithm in \(\ln(e^{-2})\) is positive, because \(e^{-2}>0\). Apply \(\ln(e^x)=x\) with \(x=-2\):
Check by exponentiating the result: \(e^{-2}\) is exactly the original logarithm argument.
Next, \(5>0\), so \(\ln(5)\) is defined. Apply \(e^{\ln(x)}=x\) with \(x=5\):
The result matches the number inside the logarithm. Finally, \(1=e^0\), so
Check: \(e^0=1\), as required by the definition of \(\ln(1)\).
Natural Logarithms and Logarithm Laws
The product, quotient, and power rules for logarithms apply to natural logarithms as they do to other valid logarithm bases. For positive \(a\) and \(b\), and real \(r\), the rules are
The positive-input conditions matter. In the product and quotient rules, \(a\) and \(b\) must be positive so that each logarithm is defined. In the power rule as stated, \(a>0\), which ensures \(a^r\) is positive for every real \(r\). These are the familiar logarithm laws with the base specified as \(e\).
The inverse identities are especially useful when an expression contains both \(e\) and \(\ln\). They do not mean that \(e\) and \(\ln\) cancel in every arrangement. The functions must be composed in the appropriate order, and the input must meet the domain condition.
How Natural Logarithms Differ from Base-10 Logarithms
A base-10 logarithm asks for the exponent on \(10\), not on \(e\). Write it explicitly as \(\log_{10}(x)\). For instance, \(\log_{10}(100)=2\), because \(10^2=100\). By contrast, \(\ln(100)\) asks for the exponent on \(e\) that gives \(100\), and its value is not \(2\).
For any positive \(x\), the change-of-base relationship connects the two logarithms:
To see why, let \(y=\log_{10}(x)\). By the definition of a logarithm, \(10^y=x\). Take the natural logarithm of both sides and use the power rule:
This argument requires \(x>0\), so that both logarithms are defined. The constant \(\ln(10)\) is approximately \(2.302585\). Thus, when the base-10 logarithm is known, multiplying by \(\ln(10)\) gives the natural logarithm. Conversely, \(\log_{10}(x)=\frac{\ln(x)}{\ln(10)}\).
Worked Example: Compare \(\ln(100)\) and \(\log_{10}(100)\)
Find both values and check each one using its defining exponential equation.
For the base-10 logarithm, identify the power of \(10\) that equals \(100\):
Check: raising \(10\) to the answer \(2\) gives the original argument \(100\).
For the natural logarithm, use \(100=10^2\) and the product rule:
Using \(\ln(10)\approx2.302585\), calculate
Check by substituting this approximate exponent into the defining equation: \(e^{4.605170}\approx100.000\). The approximation is consistent with the original argument. The two logarithms differ because their bases differ: \(10^2=100\), whereas \(e^{4.605170}\approx100\).
Using a Calculator Carefully
Many calculators have separate keys for \(\ln\) and for the common logarithm, which is usually \(\log_{10}\). Before entering a calculation, identify which base the question requires. If a calculator gives a decimal approximation, retain enough digits during intermediate calculations and round only the final result.
The inverse identities also provide a useful check on calculator work. If a calculator gives an approximation \(c\) for \(\ln(x)\), then \(e^c\) should be approximately \(x\). If it gives \(d\) for \(\log_{10}(x)\), then \(10^d\) should be approximately \(x\). Small discrepancies can occur because displayed decimals are rounded.
Worked Example: Convert a Base-10 Logarithm to a Natural Logarithm
Suppose a calculator gives \(\log_{10}(4)\approx0.602060\). Estimate \(\ln(4)\), rounding the final answer to six decimal places, and check it.
Use the change-of-base relationship:
Substitute \(\ln(10)\approx2.302585\) and the given base-10 logarithm:
Rounded to six decimal places, the estimate is \(1.386294\). To check, raise \(e\) to that exponent:
The check recovers the original positive input, \(4\), to the displayed precision.
A Common Mistake: Treating \(\ln\) as Base 10
A frequent error is to use a familiar base-10 value when the expression is a natural logarithm. For example, the following calculation is wrong:
The reasoning identifies \(\log_{10}(100)\), not \(\ln(100)\). The base in \(\ln\) is \(e\), so the correction is
The defining check makes the distinction clear: \(e^2\approx7.389\), not \(100\), while \(e^{4.605170}\approx100\). When deciding which logarithm to use, always ask which base appears in the defining exponential equation.
Why the Distinction Is Useful
Natural logarithms and base-10 logarithms both describe exponents, but their numerical values differ because their bases differ. A base-10 logarithm is convenient when powers of \(10\) are involved; the natural logarithm is convenient when powers of \(e\) are involved. Recognizing the base prevents errors in exact values, calculator work, and later calculations with exponential expressions.
For this topic, keep three ideas together: \(\ln(x)=\log_e(x)\), the inverse identities undo \(e^x\) and \(\ln(x)\) on their proper domains, and a base-10 logarithm is not a natural logarithm. These facts are enough to interpret and check the common calculations involving \(e\) and \(\ln\).
Check Your Understanding
Answer each question, paying attention to the base and to the domain of the logarithm.
- Write \(\ln(7)\) in logarithm notation that displays its base. What question does it ask?
- Evaluate \(\ln(e^4)\) and state why the inverse identity applies.
- Evaluate \(\ln(1)\), then check your result by raising \(e\) to that value.
- Find \(\log_{10}(1000)\) and explain why it is not the same calculation as \(\ln(1000)\).
- If \(\log_{10}(x)=3\), express \(\ln(x)\) in terms of \(\ln(10)\), then give its approximate value.
