The Logarithm Calculator evaluates the logarithmic power to which a base must be raised to yield a given positive real number.

### Mathematical Definition & Change of Base Formula For a positive base $b > 0$ ($b \ne 1$) and positive argument $x > 0$, the logarithm $\log_b(x) = y$ is mathematically equivalent to the exponential equation $b^y = x$: - **Common Logarithm ($\log_{10}$)**: Standard base-10 logarithm used in acoustic decibels (dB), earthquake Richter scales, and chemical pH measurements. - **Natural Logarithm ($\ln$ / $\log_e$)**: Base-$e$ logarithm ($e \approx 2.71828$) fundamental to continuous compound interest, calculus derivatives, and radioactive decay models. - **Binary Logarithm ($\log_2$)**: Base-2 logarithm standard in computer science, bit entropy calculations, and binary tree search depths. - **Change of Base Formula**: Evaluates arbitrary custom bases $b$ using natural logarithms: $$\log_b(x) = \frac{\ln(x)}{\ln(b)}$$

### Worked Calculation Example Consider finding the logarithm of **$x = 100$** with base **$b = 10$**: - **Domain Check**: $x = 100 > 0$, $b = 10 > 0$, and $b \ne 1$ (valid domain). - **Change of Base Math**: $$\log_{10}(100) = \frac{\ln(100)}{\ln(10)} = \frac{4.605170}{2.302585} = 2.0000$$ - **Natural Log Value**: $\ln(100) \approx 4.6052$ - **Binary Log Value**: $\log_2(100) = \ln(100) / \ln(2) = 4.605170 / 0.693147 \approx 6.6439$

### In-Browser Algebraic Computation All logarithmic evaluations and domain constraint checks execute client-side using JavaScript Math built-ins. No numeric values, statistical data, or formula parameters are sent across the web.