Exponential Function

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$$f\left(x\right)=a^x$$

Exponential Graphs

Practice Questions

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Change of base

$$y=a^x$$ $$lny=ln{a}^x$$ $$lny=xlna$$ $$y={\mathrm{e}}^{xlna}$$ $$a^x={\mathrm{e}}^{xlna}$$

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Follow on material:

Logarithms

Linearisation

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Linearisation of exponentials of the type:

$$Y=a{\mathrm{e}}^{bX}$$

The procedure starts with taking natural logs of the above

$$lnY=lna{\mathrm{e}}^{bX}$$

Using the addition rule for logarithms

$$lnY=lna+ln{\mathrm{e}}^{bX}$$

Using the power rule for logarithms

$$lnY=lna+bX$$ $$lnY=bX+lna$$

This is of the form

$$y=mx+c$$

as required.

This idea has been presented in terms of logs to base $e$, but can be extended to any base.

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Questions:

Year 1 Pure - Reduction to Linear Form Exam Questions
(Care of alevelmathsrevision.com)

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Further Reading:

Exponential Discharge in a Capacitor
(care of www.savemyexams.co.uk)

Nuclear Decay: Exponential and Loarithmic Equations
(care of spark.iop.org)

Absorption of Gamma Radiation
(care of www.physicsandmathstutor.com)

Linear Regression

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Logarithms.

Definition:

$$y={\mathrm{a}}^x\iff {log}_{a}y=x$$

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Rules for Logarithms

See List of logarithmic identities for more
(Care of wikipedia.org)

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Questions:

1. The curve with equation meets the curve with equation at the point $P$. Find, using algebra, the exact solution $x$ coordinate of $P$.

   Solution:

   $$3\cdot 2^x=15-2^{x+1}$$ $$3\cdot 2^x+2\cdot 2^x=15$$ $$5\cdot 2^x=15$$ $$2^x=3$$ $${log}_2{2}^x={log}_{2}3$$ $$x={log}_{2}3$$ (Alternatively from $$2^x=3$$ $$ln{2}^x=ln3$$ $$xln2=ln3$$ $$x=\frac{ln3}{ln2}$$)

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other question sources

Exponential Questions Exponentials and logarithms
(Care of www.physicsandmathstutor.com)

Logarithms
(Care of math-quiz.co.uk)

Logs Exam Questions
(Care of studywell.com)

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