Logarithms Cheatsheet

Cheatsheet Content

Definition The logarithm of a number $x$ to a base $b$ is the exponent to which $b$ must be raised to produce $x$. If $b^y = x$, then $\log_b(x) = y$. Base: $b > 0$, $b \neq 1$ Argument: $x > 0$ Result: $y$ can be any real number. Special Logarithms Common Logarithm: Base 10, denoted as $\log(x)$ or $\log_{10}(x)$. $\log(100) = 2$ because $10^2 = 100$. Natural Logarithm: Base $e$ (Euler's number, approx. $2.71828$), denoted as $\ln(x)$. $\ln(e^3) = 3$ because $e^3 = e^3$. Binary Logarithm: Base 2, denoted as $\log_2(x)$. $\log_2(8) = 3$ because $2^3 = 8$. Logarithm Rules (Properties) Product Rule: $\log_b(xy) = \log_b(x) + \log_b(y)$ Example: $\log_2(4 \cdot 8) = \log_2(4) + \log_2(8) = 2 + 3 = 5$ Quotient Rule: $\log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y)$ Example: $\log_3\left(\frac{27}{9}\right) = \log_3(27) - \log_3(9) = 3 - 2 = 1$ Power Rule: $\log_b(x^k) = k \cdot \log_b(x)$ Example: $\log_5(25^3) = 3 \cdot \log_5(25) = 3 \cdot 2 = 6$ Change of Base Formula: $\log_b(x) = \frac{\log_c(x)}{\log_c(b)}$ Commonly used to convert to natural or common log: $\log_b(x) = \frac{\ln(x)}{\ln(b)}$ $\log_b(x) = \frac{\log(x)}{\log(b)}$ Example: $\log_4(64) = \frac{\log_2(64)}{\log_2(4)} = \frac{6}{2} = 3$ Important Identities $\log_b(b) = 1$ (because $b^1 = b$) $\log_b(1) = 0$ (because $b^0 = 1$) $\log_b(b^x) = x$ $b^{\log_b(x)} = x$ $\log_b\left(\frac{1}{x}\right) = -\log_b(x)$ $\log_{b^k}(x) = \frac{1}{k} \log_b(x)$ Graphs of Logarithmic Functions For $y = \log_b(x)$: Domain: $(0, \infty)$ Range: $(-\infty, \infty)$ X-intercept: $(1, 0)$ Vertical asymptote: $x=0$ (the y-axis) If $b > 1$, the function is increasing. If $0 Example Graph ($y = \ln(x)$) y x 0 1 2 3 x y Solving Logarithmic Equations Isolate the Logarithm: Get $\log_b(x)$ by itself on one side. Convert to Exponential Form: Use the definition $b^y = x$. Example: If $\log_2(x+1) = 3$, then $2^3 = x+1 \Rightarrow 8 = x+1 \Rightarrow x=7$. One-to-One Property: If $\log_b(x) = \log_b(y)$, then $x=y$. Example: If $\log(x^2) = \log(9)$, then $x^2=9 \Rightarrow x=\pm 3$. (Check domain: $x=3$ is valid, $x=-3$ is NOT for $\log(x^2)$ if $x^2$ is not the original argument, but rather $\log(x)$ for example) Always check solutions against the domain of the original logarithmic expressions. Relationship with Exponentials Logarithms are the inverse of exponential functions. If $f(x) = b^x$, then $f^{-1}(x) = \log_b(x)$. The graphs of $y=b^x$ and $y=\log_b(x)$ are reflections across the line $y=x$. Applications pH Scale: $\text{pH} = -\log_{10}[\text{H}^+]$ Richter Scale: $M = \log_{10}\left(\frac{I}{I_0}\right)$ Decibel Scale: $\beta = 10 \log_{10}\left(\frac{I}{I_0}\right)$ Compound Interest: $A = P(1+r/n)^{nt}$ (used to solve for $t$) Population Growth/Decay: $N(t) = N_0 e^{kt}$ (used to solve for $t$)