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Precalculus JumpStart

Mathematics Department

Section 8.4 The Natural Logarithm

The base \(e\) situation is a convenient choice of base in calculus and deserves its own notation.
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Definition 8.4.1. Natural Logarithm Function.

The base \(e\) exponential function \(f(x) = e^x\) is one-to-one has an inverse function \(f^{-1}\) called the natural logarithm function denoted by \(f^{-1}(y) = \ln(y) = \log_e (y)\text{.}\)
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Thus, the notation “\(\ln\)” is just shorthand for “\(\log_e\)”. As for any other logarithm, the inverse properties are
  1. \(\displaystyle \ln(e^x) = x\text{,}\) for all \(x\text{.}\)
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  2. \(\displaystyle e^{\ln(y)} = y\text{,}\) for all \(y \gt 0\text{.}\)
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Alternatively, exponential equations are equivalent logarithmic equations according to
\begin{equation*} y = e^x \quad \text{if and only if} \quad \ln(y) =x. \end{equation*}
Observe that the natural logarithm returns the exponent in the base \(e\) exponential equation. Below is the graph of the natural logarithm (solid curve).
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Figure 8.4.2.
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Example 8.4.3. Using Technology.

According to my calculator,
\begin{equation*} \ln(10) \approx 2.303. \end{equation*}
Interpreting this in exponential form, we find
\begin{equation*} e^{2.303} \approx 10. \end{equation*}
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Example 8.4.4. Evaluating Logarithms.

Evaluate the following logarithms without the use of calculator.
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  1. \(\displaystyle \displaystyle \ln \left(\sqrt[3]{e}\right)\)
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  2. \(\displaystyle \displaystyle \ln \left(\frac{1}{e^3}\right)\)
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  3. \(\displaystyle \displaystyle \ln \left(e\right)\)
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  4. \(\displaystyle \displaystyle \ln \left(1\right)\)
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  5. \(\displaystyle \displaystyle \ln \left(0\right)\)
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Solution.
  1. \(\displaystyle e^\boxed{1/3} = \sqrt[3]{e} \quad\Rightarrow\quad \ln \left(\sqrt[3]{e}\right) = \boxed{1/3}\)
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  2. \(\displaystyle e^{\boxed{-3}} = \frac{1}{e^3} \quad\Rightarrow\quad \ln \left(\frac{1}{e^3}\right) = \boxed{-3}\)
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  3. \(\displaystyle e^{\boxed{1}} = e \quad\Rightarrow\quad \ln \left(e\right) = \boxed{1}\)
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  4. \(\displaystyle e^{\boxed{0}} = 1 \quad\Rightarrow\quad \ln \left(1\right) = \boxed{0}\)
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  5. \(e^x \gt 0\text{,}\) for all \(x\text{.}\) In particular, \(e^x \neq 0\text{,}\) for any \(x\text{.}\) Thus \(\ln \left(0\right)\) is undefined.
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Example 8.4.5. Graphing Logarithmic Functions.

Find the domain and sketch the graph of \(y = \ln(x+1)\text{.}\)
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Solution.
We require that the input into the logarithm is positive. In this case, \(x+1 \gt 0\) or equivalently, \(x \gt -1\text{.}\) The domain is then the interval \((-1,+\infty)\text{.}\) Adding one to the input has the effect of translating the graph one unit to the left so that the vertical asymptote of the logarithm is now \(x=-1\text{.}\)
Figure 8.4.6. The graph of \(y = \ln(x+1)\) (solid) compared to \(y = \ln(x)\) (dashed).
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