**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= log_{3}x

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 log _{a}x**

\(a^y\) = x => y = log_{a}x

**– Laws of Logarithms**

y=log_{a}x => x = \(a^y\)

log_{a}x^{n }= n log_{a}x

log_{a}xy = log_{a}x + log_{a}y

log_{a}\(\frac{x}{y}\) = log_{a}x + log_{a}y

log_{a}1 = 0

log_{a}a = 1

The natural log is often written In

log_{a}b = \(\frac{log_cb}{log_ca }\) (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!!!

log_{a}a^{x }= log_{a}b => x = log_{a}b

**– 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) \(e^{-γt}\)