Before You Start Exponential functions Equation solving Calculator use for powers What You Will Learn Explain a logarithm as the answer to an exponent question. Use logarithm laws to rewrite expressions. Solve exponential and logarithmic equations while checking domains. Preview Activity: How Long Until Half Remains? A battery starts at 80\% and loses half of its remaining charge every 6 hours. A simple model is B(t)=80(0.5)^{t/6}. 1. Find B(0), B(6), and B(12). 2. Estimate when the battery reaches 20\%. 3. Explain why solving 20=80(0.5)^{t/6} requires undoing an exponent. Hint: Divide both sides by 80 first. Then ask: what exponent makes 0.5 equal to 0.25? Solution: B(0)=80, B(6)=40, and B(12)=20, so the battery reaches 20\% after 12 hours. The Logarithm \logb x=y \quad \text{if and only if} \quad b^y=x The logarithm \logb x asks: to what power must b be raised to get x? Because logarithms undo exponentials, they are the natural tool when the unknown is in the exponent. Exponential and Logarithm as Inverses The graphs of y=e^x and y=\ln x mirror each other across y=x. Core Logarithm Laws \logb(xy)=\logb x+\logb y turns multiplication into addition. \logb\left(\frac{x}{y}\right)=\logb x\logb y turns division into subtraction. \logb(x^r)=r\logb x moves powers down as factors. \ln(e^x)=x and e^{\ln x}=x show the inverse relationship. Worked Example: Solve an Exponential Equation Solve 3^x=17. Solution: Take natural logs on both sides: \ln(3^x)=\ln(17). Use the power law: x\ln 3=\ln 17. Therefore x=\frac{\ln 17}{\ln 3}\approx 2.58. Activity: Expand and Condense 1. Expand \log(x^2y^5). 2. Condense \ln 4+3\ln x\ln(y+1) into one logarithm. 3. Solve \ln(x2)=\ln 7 and state the domain restriction. Hint: Use the product, quotient, and power laws. For the last question, x2 must be positive. Solution: \log(x^2y^5)=2\log x+5\log y. \ln 4+3\ln x\ln(y+1)=\ln\left(\frac{4x^3}{y+1}\right). x2=7, so x=9, and the original equation requires x2. Summary A logarithm is an exponent written from the opposite direction. Log laws are exponent laws translated into logarithm language. Always check that every logarithm input is positive. Try These 1. Rewrite 2^5=32 as a logarithmic statement. 2. Solve 5e^{0.2t}=30. 3. Find the domain of f(x)=\ln(2x6).