Logarithmic Functions Summary

Logarithmic Functions

Converting Between Logarithmic and Exponential Forms

• $log_{b}(x) = y$ is equivalent to $b^{y} = x$, where b > 0 and $b \neq 1$.
• $y = log_{b}(x)$ and $y = b^{x}$ are inverse functions.
• For x > 0, b > 0, and $b \neq 1$, $log_{b}(x) = y$ is equivalent to $b^{y} = x$.

Domain and Range

• For exponential functions:
  • Domain: All real numbers.
  • Range: $(0, \infty)$
• For logarithmic functions:
  • Domain: $(0, \infty)$
  • Range: $(-\infty, \infty)$

Common Logarithms

• Base 10. Denoted as $y = log(x)$, which means $log_{10}(x)$.
• Equivalent to $x = 10^{y}$.

Natural Logarithms

• Base $e$. Denoted as $y = ln(x)$, which means $log_{e}(x)$.
• Equivalent to $x = e^{y}$.

Examples of Conversion

• $log_{c}(4) = 10$ in exponential form: $c^{10} = 4$
• $log_{2}(9) = x$ in exponential form: $2^{x} = 9$
• $log(d) = f$ in exponential form: $10^{f} = d$
• $ln(f) = h$ in exponential form: $e^{h} = f$
• $4^{x} = y$ in logarithmic form: $log_{4}(y) = x$
• $10^{a} = n$ in logarithmic form: $log(n) = a$
• $e^{2} = b$ in logarithmic form: $ln(b) = 2$
• $n^{4} = 103$ in logarithmic form: $log_{n}(103) = 4$

Evaluating Logarithms Mentally

• Rewrite the argument $x$ as a power of $b$, i.e., $b^{y} = x$
• Determine the exponent $y$ that $b$ must be raised to in order to get $x$.

Examples of Evaluation without a Calculator

• $log_{3}(\frac{1}{27})$:
  • $3^{y} = \frac{1}{27}$
  • $y = -3$
• $log_{2}(\frac{1}{8}) + 4$:
  • $2^{y} = \frac{1}{8}$
  • $y = -3$
  • $-3 + 4 = 1$
• $log(10,000)$:
  • $10^{y} = 10,000$
  • $y = 4$
• $log(1) + 7$:
  • $10^{y} = 1$
  • $y = 0$
  • $0 + 7 = 7$
• $ln(e^{1})$:
  • $e^{y} = e^{1}$
  • $y = 1$

Evaluating Logarithms Using a Calculator

• $log(14.5) \approx 1.1614$
• $ln(3.28) \approx 1.1878$

Real World Example

• Exposure Index: $EI = log_{2}(\frac{f^{2}}{t})$, where $f$ is the f-stop setting and $t$ is the exposure time.
• Given $f = 8$ and $t = 2$, $EI = log<em>{2}(\frac{8^{2}}{2}) = log</em>{2}(32)$.
• $2^{y} = 32$, so $y = 5$. Therefore, $EI = 5$.