6.Exponentials and Logarithms
– Function of y=\(a^x\)and y=\(e^x\)
y= \(3^x\) is an exponential function
And the inverse is the logarithmic function
y= log3x
A function and its inverse are the reflections in the line
– Gradient of exponential functions
y = \(a^x\) =>\(\frac{dy}{dx}\) = \(a^x In~a\)
– Definition of logax
\(a^y\) = x => y = logax
– Laws of Logarithms
y=logax => x = \(a^y\)
logaxn = n logax
logaxy = logax + logay
loga\(\frac{x}{y}\) = logax + logay
loga1 = 0
logaa = 1
The natural log is often written In
logab = \(\frac{log_cb}{log_ca }\) (Changing the base of a logarithm)
– Solving equation \(a^x\) =b
\(a^x\) =b
In(\(a^x\)) = In b (by taking natural logs)
x ln(a)=lnb
x = \(\frac{In b}{In a}\)
You can take any base!!!
logaax = logab => x = logab
– Logarithmic Graphs
Graphs of y = \(e^x\) and y = In x are inverse functions and are reflections in the line y = x
Graph of y = \(e^{ax+b}\) + c
The graph of y = \(e^{2x}\) is the graph of y = \(e^x\) stretched by a factor of 0.5 in direction of x-axis.
The graph of y = \(e^{2x+3}\) is the graph of y = \(e^{2x}\) translated through (-3/2, 0) and the graph of y = \(e^{2x+3}\) +4 is the graph of y = \(e^{2x+3}\) translated through (0,4)
– Application of exponential Growths and Decay
Solving \(e^{ax+b}\) =p
Taking naural logs on both sides
ax + b = In p => x = \(\frac{In p - b}{b}\)
Radioactive decay is – \(\frac{dN}{dt}\) = γN(t) and the solution is
N(t) = N(0) \(e^{-γt}\)