Decadic (base-10) Logarithm of Complex Numbers

Calculation of \(\log_{10}(z)\) - logarithm to base 10

Log₁₀ Calculator

Decadic Logarithm \(\log_{10}(z)\)

The decadic logarithm (base 10) of a complex number is computed by converting the natural logarithm: \(\log_{10}(z) = \frac{\ln(z)}{\ln(10)}\)

Complex number z = a + bi
+
i
Calculation Result
\(\log_{10}(z)\) =
Conversion: \(\log_{10}(z) = \frac{\ln(z)}{\ln(10)} = \frac{\ln(z)}{2.302585}\)

Log₁₀ - Properties

Base Conversion
\[\log_{10}(z) = \frac{\ln(z)}{\ln(10)}\]

Conversion from natural to decadic logarithm

Formula
\[\log_{10}(z) = \frac{\log_{10}|z| + i\arg(z)}{\ln(10)}\]
Base 10
ln(10) ≈ 2.303
Important Properties
  • \(\log_{10}(z_1 \cdot z_2) = \log_{10}(z_1) + \log_{10}(z_2)\)
  • \(\log_{10}(z_1 / z_2) = \log_{10}(z_1) - \log_{10}(z_2)\)
  • \(\log_{10}(z^n) = n\log_{10}(z)\)
  • \(10^{\log_{10}(z)} = z\)
Special Values
  • \(\log_{10}(10) = 1\)
  • \(\log_{10}(100) = 2\)
  • \(\log_{10}(1000) = 3\)
  • \(\log_{10}(1) = 0\)
  • \(\log_{10}(0.1) = -1\)
Conversion

From ln to log₁₀: \(\log_{10}(z) = \frac{\ln(z)}{2.302585}\)
From log₁₀ to ln: \(\ln(z) = 2.302585 \cdot \log_{10}(z)\)

Formulas for the Decadic Logarithm

The decadic logarithm (base 10) of a complex number \(z = a + bi\) is computed by converting the natural logarithm.

Base conversion
\[\log_{10}(z) = \frac{\ln(z)}{\ln(10)}\]

With \(\ln(10) \approx 2.302585\)

Detailed form
\[\log_{10}(z) = \frac{\ln|z| + i\arg(z)}{\ln(10)}\]

With real and imaginary parts separated

Calculation Example

Calculation: \(\log_{10}(100 + 0i)\) and \(\log_{10}(3 + 4i)\)
Example 1: Real number \(\log_{10}(100)\)

Given: \(z = 100 + 0i\)

Computation:

\(\log_{10}(100) = \frac{\ln(100)}{\ln(10)}\)

\(= \frac{4.605}{2.303} = 2\)

Result: \(2 + 0i\)

As in the real case: \(10^2 = 100\) ✓

More examples

\(\log_{10}(10) = 1\)

\(\log_{10}(1000) = 3\)

\(\log_{10}(1) = 0\)

\(\log_{10}(0.1) = -1\)

Example 2: Complex number \(\log_{10}(3+4i)\)

Step 1: Natural logarithm

\(|z| = \sqrt{3^2+4^2} = 5\)

\(\arg(z) = \arctan(4/3) \approx 0.927\) rad

\(\ln(z) = \ln(5) + 0.927i \approx 1.609 + 0.927i\)

Step 2: Convert to base 10

\(\log_{10}(z) = \frac{1.609 + 0.927i}{2.303}\)

Result: \(0.699 + 0.403i\)

Verification

Check: \(10^{0.699+0.403i}\)
\(= 10^{0.699} \cdot 10^{0.403i}\)
\(= 5.0 \cdot (\cos(0.927) + i\sin(0.927))\)
\(\approx 3.0 + 4.0i\) ✓

Component-wise computation
Real part:
\[\text{Re}(\log_{10}(z)) = \frac{\ln|z|}{\ln(10)} = \frac{\frac{1}{2}\ln(a^2+b^2)}{\ln(10)}\]
Imaginary part:
\[\text{Im}(\log_{10}(z)) = \frac{\arg(z)}{\ln(10)} = \frac{\arctan(b/a)}{\ln(10)}\]

Comparison: log₁₀ vs. ln vs. log₂

Natural logarithm (ln)

Base: \(e \approx 2.718\)

Formula: \(\ln(z)\)

Use: Analysis, mathematics

Example: \(\ln(e) = 1\)

Decadic logarithm (log₁₀)

Base: \(10\)

Formula: \(\log_{10}(z) = \frac{\ln(z)}{\ln(10)}\)

Use: Physics, engineering, pH-scale

Example: \(\log_{10}(100) = 2\)

Binary logarithm (log₂)

Base: \(2\)

Formula: \(\log_2(z) = \frac{\ln(z)}{\ln(2)}\)

Use: Computer science, bits

Example: \(\log_2(8) = 3\)

Conversion factors

\(\ln(10) \approx 2.303\)

Factor for ln → log₁₀

\(\ln(2) \approx 0.693\)

Factor for ln → log₂

\(\frac{\ln(10)}{\ln(2)} \approx 3.322\)

Factor for log₂ → log₁₀

Applications of the Decadic Logarithm

Natural Sciences

  • pH value: \(\text{pH} = -\log_{10}[\text{H}^+]\)
  • Decibel (dB): \(L = 20\log_{10}\left(\frac{p}{p_0}\right)\)
  • Richter scale: earthquake magnitude
  • Stellar brightness: magnitude

Engineering

  • Bode plot: gain in dB
  • Signal processing: dynamic range
  • Acoustics: sound pressure level
  • Electrical engineering: attenuation and gain

Practical Examples

Example: pH value

If \([\text{H}^+] = 10^{-7}\) mol/L
\(\text{pH} = -\log_{10}(10^{-7}) = 7\) (neutral)

Example: Decibels

Gain by factor 100:
\(20\log_{10}(100) = 20 \cdot 2 = 40\) dB

Example: Richter scale

Magnitude 5 is 10 times stronger than magnitude 4:
\(\log_{10}(10) = 1\) unit difference

Important note

Notation: In many calculators and programming languages "\(\log\)" denotes \(\log_{10}\) while "\(\ln\)" denotes the natural logarithm. In mathematics, "\(\log\)" often refers to the natural logarithm!


More complex functions

Absolute value (abs) • Angle • Conjugate • Division • Exponent • Logarithm to base 10 • Multiplication • Natural logarithm • Polarform • Power • Root • Reciprocal • Square root •
Cosh • Sinh • Tanh •
Acos • Asin • Atan • Cos • Sin • Tan •
Airy function • Derivative Airy function •
Bessel-I • Bessel-Ie • Bessel-J • Bessel-Je • Bessel-K • Bessel-Ke • Bessel-Y • Bessel-Ye •