Exponential and logarithmic functions are inverse ideas. They model fast growth, rapid decay, and scale changes that linear models cannot describe well.
An exponential function has variable in the exponent: \[ y=a^x,\quad a>0,\ a\ne1. \]
If \(a>1\), the function grows as \(x\) increases. If \(0<a<1\), it decays.
Example
\[ 2^x \text{ grows},\qquad \left(\frac12\right)^x \text{ decays}. \]
A logarithm asks for the exponent: \[ \log_a b = c \iff a^c=b. \]
So \(\log_2 8=3\) because \(2^3=8\). Also, \(\log_{10}100=2\).
These two graphs are reflections of each other in the line \(y=x\).
Exponential solving often uses:
\[ a^m a^n=a^{m+n},\quad \frac{a^m}{a^n}=a^{m-n},\quad (a^m)^n=a^{mn}. \]
For \(M,N>0\):
\[ \log_a(MN)=\log_a M+\log_a N, \] \[ \log_a\!\left(\frac{M}{N}\right)=\log_a M-\log_a N, \] \[ \log_a(M^k)=k\log_a M. \]
Common mistakes: \(\log(M+N)\ne \log M+\log N\).
To evaluate logs with calculator: \[ \log_a b = \frac{\log b}{\log a} = \frac{\ln b}{\ln a}. \]
Here \(\log\) usually means base 10 and \(\ln\) means base \(e\).
Case 1: same base.
\[ 3^{2x-1}=3^5 \Rightarrow 2x-1=5 \Rightarrow x=3. \]
Case 2: take logs.
\[ 5^x=17 \Rightarrow x=\log_5 17=\frac{\log 17}{\log 5}\approx 1.760. \]
Example 1. \(\log_2(x-1)=4\)
Convert to exponential: \(x-1=2^4=16\), so \(x=17\). Domain check: \(x>1\), valid.
Example 2. \(\log_{10}(x+3)+\log_{10}(x-1)=1\)
\[ \log_{10}\big((x+3)(x-1)\big)=1 \Rightarrow (x+3)(x-1)=10. \] \[ x^2+2x-3=10 \Rightarrow x^2+2x-13=0. \] \[ x=-1\pm\sqrt{14}. \] Domain requires \(x>1\), so \(x=-1+\sqrt{14}\) only.
A common model: \[ A(t)=A_0(1+r)^t \] for growth (\(r>0\)) or decay (\(r<0\), with \(1+r>0\)).
Continuous model: \[ A(t)=A_0e^{kt}, \] where \(k>0\) gives growth and \(k<0\) gives decay.