Exponential and Logarithmic Functions
Exponential and Logarithmic Functions
Exponential and logarithmic functions are inverses, and nearly every question in this domain rests on that relationship plus the laws of logarithms.
Concepts. For $f(x)=a\cdot b^x$ with $b>0$, $b\ne1$, the graph has a horizontal asymptote $y=0$, a $y$-intercept at $a$, and shows growth if $b>1$ or decay if $0<b<1$. The inverse is $y=\log_b x$, defined only for $x>0$, with a vertical asymptote at $x=0$. The change-of-base formula $\log_b x=\dfrac{\log x}{\log b}$ lets any calculator evaluate any base. The three laws - product, quotient, and power - drive equation solving: isolate the exponential, take a logarithm of both sides, or condense logarithms and convert to exponential form. Applications include compound interest $A=P(1+i)^n$, continuous growth $A=Pe^{rt}$, half-life, and exponential regression.
Traps. $\log(x+y)\ne\log x+\log y$; the product law applies to a product, not a sum. When solving $\log_b M=\log_b N$ you may equate arguments, but you must reject any solution that makes an original argument non-positive. Students also forget that $\ln$ is $\log_e$ and mix bases mid-problem. Watch the coefficient: $2\log x$ becomes $\log x^2$, not $(\log x)^2$.
Rule of thumb. To solve for an unknown exponent, take logs; to solve for an unknown inside a log, rewrite in exponential form. Always check the domain at the end.
Worked example. Solve $3^{2x-1}=20$. Take $\log$ of both sides: $(2x-1)\log 3=\log 20$, so $2x-1=\dfrac{\log 20}{\log 3}\approx2.727$, giving $x\approx1.86$.