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.
Logarithms were developed in the early 17th century (notably by John Napier) to simplify long multiplication and division into addition and subtraction. Before electronic calculators, this was a major speed boost for astronomy, navigation, and engineering.
Exponential functions became central in science through models of population growth, radioactive decay, and continuously changing quantities.
Let \(\log_a M=p\) and \(\log_a N=q\). Then \(M=a^p\), \(N=a^q\), so \[ MN=a^{p+q}. \] Taking \(\log_a\) gives: \[ \log_a(MN)=p+q=\log_a M+\log_a N. \]
If \(y=a^x\), swapping \(x,y\) gives \(x=a^y\), so \(y=\log_a x\). That is why exponential and logarithmic graphs are mirror images across \(y=x\).
Answer: \(\log_3 81=4\), \(\log_{10}0.01=-2\).
Answer: \(\log_2 8 + \log_2 x^3 = 3 + 3\log_2 x\).
Answer: \(x+1=\log_2 7\), so \(x=\log_2 7-1 \approx 1.807\).
Domain: \(x-4>0\Rightarrow x>4\).
Answer: \(x-4=25\Rightarrow x=29\), valid.
Model: \(A(t)=500(1.06)^t\).
Value: \(A(4)=500(1.06)^4\approx 631.24\), about 631.
Answer: Decay, because exponent coefficient is negative (\(k=-0.18\)).
Savings and loans often follow exponential rules. Logarithms are used to solve for unknown time or interest rate.
Radioactive substances and medicine concentration in the body are modeled with exponential decay.
pH, decibels, and Richter magnitude use logarithmic scales to represent very large ratio ranges in manageable numbers.