# Logarithm Rules Logarithm is the [[Inverse (in Algebra)|inverse]] of exponentiation. For exponentiation rules see [[Exponent Rules]]. For the logarithmic scale see [[Logarithmic Scale]]. By definition, for every $b \ne 1$: $b^{\log_b a} = a$ ^hmy4h0 $ \begin{align} \log_x1 &= 0 \\ \\ \log(ab) &= \log a + \log b \\ \\ \log\left(\frac a b \right) &= \log a - \log b \\ \\ \log(a^n) &= n \cdot \log a \\ \\ \log_ba &= \frac{1}{\log_ab}\\ \\ \log_b a &= \frac{\log_c a}{\log_c b} \\ \\ \log_c b \cdot \log_b a &= \log_c a \end{align} $ The last rule can be derived from the change of base formula: $ \begin{align} \log_c b \cdot \log_b a &= \log_c b \cdot \frac{\log_c a}{\log_c b} & \text {change the base of the second term} \\ &=\log_c a \end{align} $ ## Deriving The Change of Base Formula $ \begin {align} \log_b a &= y \\ b^y &= a \\ \log_c b^y &= log_c a \\ y \log_c b &= log_c a \\ y &= \frac {log_c a}{\log_c b} \end {align} $ ## Logarithm Derivative $\frac d {dx} \bigg[\log_b x\bigg] = \frac 1 {x \ln b}$ $\frac d {dx} \bigg[\ln x\bigg] = \frac 1 x$^kl76b8 ### Proof ![[Logarithm Rules#^hmy4h0]] If we take the derivative of both sides and apply [[The Derivative Chain Rule]] we get: $ \begin{gather} \frac d {dx}\bigg [ b^{\log_b x} \bigg] = 1 \\ b^{\log_b x} \ln b \cdot \frac d {dx} \bigg[\log_bx\bigg] = 1 &\text{see note 1} \\ \frac d {dx} \bigg[\log_bx\bigg] = \frac 1 {x\ln b} \end{gather} $ Note 1: see [[Exponential Functions#Exponential Function Derivative]] ![[Exponential Functions#^x4xm54]] ## Logarithm Antiderivative $\int \ln (nx) \, dx = x(\ln (nx) - 1) + C$ ^scpwa7 ### Proof For $\int \ln (x) \,dx$, apply [[Integration by Parts]] with $u = \ln x$ $\int \ln x \, dx = (\ln(x)) (x) - \int \left(\frac 1 x\right) (x) \, dx = x(\ln(x) - 1) + C$ $ \begin {align} \int \ln(nx) \, dx &= \int \ln x \, dx + \int \ln n \, dx \\ &= x\ln x - x + x\ln n + C \\ &= x(\ln x + \ln n -1) + C \\ &= x(\ln (nx) - 1) + C \end {align}$