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⬅ Solving exponential inequalities Transforming a relationship to linear form ➡
Logarithmic and exponential functions Solving logarithmic equations Solving exponential equations Solving exponential inequalities Natural logarithms Transforming a relationship to linear form

Logarithms — Natural logarithms 43 problems

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📘 Notes

Logarithms — Natural Logarithms

Natural logarithms are logarithms with base \(e\). They are written as \(\ln x\).

1. What is a natural logarithm?

The natural logarithm of a number \(x\) is the power to which \(e\) must be raised to give \(x\).

\[ \ln x=\log_e x \]

Here, \(e\) is a special irrational number:

\[ e \approx 2.71828 \]

2. Converting between logarithmic and exponential form

These two statements mean the same thing:

\[ \ln x=y \qquad \Longleftrightarrow \qquad e^y=x \]

Example:

\[ \ln 7 = y \qquad \Longleftrightarrow \qquad e^y = 7 \]

3. Key facts

\[ \ln 1=0 \qquad \text{because} \qquad e^0=1 \] \[ \ln e=1 \qquad \text{because} \qquad e^1=e \]

Also:

\[ e^{\ln x}=x \quad (x>0) \] \[ \ln(e^x)=x \]

4. Laws of natural logarithms

The ordinary logarithm laws also work for natural logarithms:

\[ \ln(ab)=\ln a+\ln b \] \[ \ln\left(\frac{a}{b}\right)=\ln a-\ln b \] \[ \ln(a^n)=n\ln a \]

These are very useful when simplifying expressions and solving equations.

5. Important restriction

A natural logarithm is only defined for positive values.

\[ \ln x \text{ exists only if } x>0 \]

For example, \(\ln 5\) exists, but \(\ln 0\) and \(\ln(-3)\) do not exist.

6. Worked examples

Example 1: Simplify

\[ \ln 4+\ln 3 \]

Use \(\ln(ab)=\ln a+\ln b\):

\[ \ln 4+\ln 3=\ln 12 \]

Example 2: Simplify

\[ 2\ln x \]

Use \(\ln(a^n)=n\ln a\):

\[ 2\ln x=\ln(x^2) \]

Example 3: Solve

\[ \ln x=2 \]

Change to exponential form:

\[ x=e^2 \]

7. Natural logarithms and exponential equations

Natural logarithms are especially useful for solving equations involving \(e\).

Example: Solve

\[ e^{3x}=10 \]

Take natural logarithms of both sides:

\[ \ln(e^{3x})=\ln 10 \] \[ 3x=\ln 10 \]

\[ x=\frac{\ln 10}{3} \]

8. Exam tips

  • Remember that \(\ln x\) means \(\log_e x\).
  • Know the facts \(\ln 1=0\) and \(\ln e=1\).
  • Use the usual logarithm laws when simplifying.
  • Always check that the argument of \(\ln\) is positive.
  • When solving equations, use \( \ln(e^x)=x \) and \( e^{\ln x}=x \).
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