Logarithms
Logarithm Solver
Logarithm Theory
A logarithm answers the question: "To what exponent must we raise a given base to obtain a specific number?" If $b^y = x$, then the logarithm of $x$ to base $b$ is $y$. This is written as: $y = \log_b(x) \iff b^y = x$. Where:
- $b$ is the base. It must be positive and not equal to 1 ($b > 0, b \neq 1$).
- $x$ is the argument. It must be positive ($x > 0$).
- $y$ is the logarithm (the exponent).
Common Logarithm: Base 10, written as $\log_{10}(x)$ or $\log(x)$.
Natural Logarithm: Base $e$ (Euler's number, $e \approx 2.71828$), written as $\log_e(x)$ or $\ln(x)$.
Assume $M > 0, N > 0, b > 0, b \neq 1$, and $P$ is any real number:
- Product Rule: $\log_b(M \cdot N) = \log_b(M) + \log_b(N)$
- Quotient Rule: $\log_b\left(\frac{M}{N}\right) = \log_b(M) - \log_b(N)$
- Power Rule: $\log_b(M^P) = P \cdot \log_b(M)$
- Change of Base: $\log_b(M) = \frac{\log_c(M)}{\log_c(b)}$
- Identities: $\log_b(b^x) = x$ and $b^{\log_b(x)} = x$ (for $x > 0$)
- Special Values: $\log_b(1) = 0$ and $\log_b(b) = 1$
Characteristics of $y = \log_b(x)$ for $b > 1$:
- Domain: $(0, \infty)$
- Range: $(-\infty, \infty)$
- X-intercept: $(1, 0)$
- Vertical Asymptote: $x=0$ (y-axis)
Graph of $y = \log_{10}(x)$
Graph of $y = \ln(x)$