# Exponentials and Logarithms

**AS level content**

- Know and use the function x
^{a}and its graph, where a is positive - Know and use the function e
^{x}and its graph - Know that the gradient of e
^{kx}is equal to kekx and hence understand why the exponential model is suitable in many applications - Know and use the definition of log
_{a}x as the inverse of a^{x}, where a is positive and x≥0 - Know and use the function lnx and its graph
- Know and use lnx as the inverse function of e
^{x} - Understand and use the laws of logarithms:

Log_{a}x + Log_{a}y = Log_{a}(xy)

Log_{a}x - Log_{a}y = Log_{a}(x/y)

kLog_{a}x = Log_{a}x^{k}

- Solve equations of the form a
^{x}= b - Use logarithmic graphs to estimate parameters in relationships of the form y=ax
^{n}and y = kb^{x}, given data for x and y - Understand and use exponential growth and decay; use in modelling

- (examples may include the use of e in continuous compound interest, radioactive decay, drug concentration decay, exponential growth as a model for population growth);
- consideration of limitations and refinements of exponential models

### The Exponential Function

This interactive resource is designed to enable students to explore the nature of the exponential function and the derivative of the exponential function.

An introduction page sets out some basic information about exponential functions leading to a definition of the exponential function. There follows an interactive graphical page showing graphically the definition of the exponential function. The next interactive graph enables students to find an approximation for the value of e. There follows a summary of the main points.

### Exponential graphs

This card-sorting activity explores exponential graphs.

Exponential graphs: Teacher guidance

This teacher guidance gives an overview of the task including prior student knowledge, suggested approaches and possible extensions.

Exponential graphs: Activity

The PowerPoint presentation requires students to match graphs into different classes and match the graphs with equations, as well as sketching the graphs of various exponential functions.

### Exponential growth and decay

This package of resources introduces students to exponential growth and decay by exploring exponential functions, compound interest for savings and depreciation of assets.

Exponential Growth and Decay: Teacher Guidance

This teacher guidance contains information including what prior student knowledge is required, suggested approaches and possible extension activities.

Exponential Growth and Decay: Changing exponential variables

This presentation, worksheet and interactive Geogebra file help students explore a series of exponential functions, and effects when changing the variables in the equation

Exponential Growth and Decay: Compound interest and Depreciation

This sequence of worksheets explore compound interest and depreciation by allowing students to complete tables and plots graphs for a number of different examples.

### Exponential and Logarithmic Functions

This resource contains a number of interactive excel spreadsheets for use as stimulation for discussion and teacher led sessions.

e

This resource shows how the value for e is derived from the binomial series for (1+x)n and how this is used to derive the series for e^{x}. It also finds the derivative of ex and explores the relationship between the total interest generated and the number of times compound interest is added to an investment. The final sheet investigates the area under the graph of 1/x and derives the integral of 1/x.

Logarithms

This resource begins with a sheet listing the values of an and the value of logn where n can be altered. The second sheet uses the base e and introduces natural logarithms. The next two sheets require students to estimate the values of logarithms to different bases from the graphs shown. The fourth sheet illustrates the exponential and logarithmic functions on one graph where different bases can be selected. Subsequent sheets deal with the sum and difference of logs, the log ofalogx, and changing the base of logarithms. The final two sheets generate questions where students need to evaluate logs and answer questions based on the laws derived in the previous sheets. Answers can be revealed and new sets of questions generated.

Exponential and Logarithmic Functions

The first two sheets explore the graphs of exponential functions. The base can be altered and the appropriate graphs shown. Negative exponents are also explored and the relationship between the graphs of ax and a-x illustrated. The third sheet explores the graph of ekx and the relationship between the value of the function and its gradient. The next two sheets show the derivation of the value of e as a limit and the sum of the series. The final sheet shows the graphs of exponential functions together with the inverse logarithmic functions.

### Logarithms

Building Log Equations requires students to form equations given a set of cards and to determine, with examples, whether the equation is always, sometimes or never true and to attempt to say why. Students must include at least one log card in their equation and will need to be familiar with logs in different number bases.

### Logarithms

This Active A level resource contains six problems which require deep understanding of the working of logarithms. Students are required to develop questions which give a specific answer, with different solutions possible. This presents an opportunity for rich discussion as students question and justify the solutions presented as do the true/false questions included in the resource.

### Simplifying Expressions Using the Rules of Logarithms

This resource contains four activities designed to practise the basics and extend understanding of logarithms. There is no need for any additional teaching before each activity – the student can work out the next stage from their existing knowledge. The package includes:

Logarithms: true or false: pairs of students are presented with twelve statements and have to decide whether each statement is true or false and provide justification.

Logarithms: ordering: students have to place cards in order of value using logarithms that do not have simple rational answers and therefore must be estimated in order to compare values.

Logarithms: odd-one-out: students have to decide which of the three statements is the odd-one-out and then write an expression to match the odd-one-out. The activity contains numerical and algebraic values.

Logarithms: matching up: students have to focus on the laws of logarithms in order to match cards with equivalent values. Not all cards pair up, some have more than one partner and some have no partner at all.

Logarithms: open questions: a series of open-ended, probing questions, designed to assess students' knowledge, identify misconceptions and inform future teaching.

### Simplifying Logarithmic Expressions A13

This resource is designed to help students to develop their understanding of the laws of logarithms, practise using the laws of logarithms to simplify numerical expressions involving logarithms and apply their knowledge of the laws of logarithms to expressions involving variables. Students should have some knowledge of the laws of logarithms as applied to numerical expressions