Year 11: Exponential and Logarithmic Functions

5 dot points across 5 inquiry questions. Click any dot point for a focused answer with worked past exam questions where available.

What do the graphs of y=a^x and y=log_a x actually look like, why are they exact mirror images of each other, where are their asymptotes, and how do translations, reflections and dilations move them while you read off the domain and range?

Graph exponential and logarithmic functions, identify their asymptotes, use the reflection in the line y=x that makes them inverse functions, apply translations, reflections and dilations, and state the domain and range of each

The Year 11 Maths Advanced answer on exponential and logarithmic graphs: the shapes of y=a^x and y=log_a x, their horizontal and vertical asymptotes, why the two curves are exact mirror images in the line y=x, how translations, reflections and dilations move them, and how to state domain and range, with code-checked diagrams and original practice questions.

20 min answer →

What do the index laws really mean once the index can be zero, negative or a fraction, and how do you simplify index expressions, solve a simple index equation and use scientific notation without a slip?

Apply the index laws to expressions with rational indices: use zero, negative and fractional indices, simplify and evaluate index expressions, solve simple index equations, and write numbers in scientific notation

A focused answer to the Year 11 Maths Advanced groundwork on indices: the five index laws for the same base, zero and negative indices as reciprocals, fractional indices as roots and powers, the reciprocal-then-root-then-power order for messy indices, solving simple index equations by matching bases, and scientific notation, with worked examples and original practice questions.

18 min answer →

What is a logarithm really, how do you switch between index form and log form, and how do the log laws let you expand, contract, evaluate and solve - including the cases a calculator only does in base 10?

Define logarithms as indices, convert between index form and logarithmic form, apply the logarithm laws (product, quotient, power), use the logarithms of 1 and of the base, change the base, and work with common logarithms

The Year 11 Maths Advanced answer on logarithms: a logarithm is the index, how to convert between index form and log form, the product, quotient and power laws, the logs of 1 and of the base, the change-of-base formula and common (base 10) logs, with code-checked worked examples and original practice questions.

19 min answer →

What is a radian, why does measuring an angle by arc length over radius make pi rad = 180 deg, what are the exact radian values of the common angles, and how do the clean formulas arc length L = r theta, sector area A = (1/2) r^2 theta and segment area (sector minus triangle) follow once an angle is measured in radians?

Define a radian as the angle whose arc length equals the radius, convert between degrees and radians using pi rad = 180 deg, know the exact radian values of the common angles, and use the radian formulas for arc length L = r theta, sector area A = (1/2) r^2 theta and the area of a segment as the sector minus the triangle

The Year 11 Maths Advanced answer on radian measure, arcs and sectors: a radian defined by arc length equal to radius, converting with pi rad = 180 deg, exact radian values of common angles, arc length L = r theta, sector area half r squared theta, and segment area as sector minus triangle, with code-checked numbers and diagrams.

18 min answer →

What is the special number e (about 2.718), why is it singled out as the base whose graph y=e^x has gradient exactly 1 where it crosses the y-axis, what is the natural logarithm ln x as its inverse, and how do you convert between e^x and ln to solve equations and transform the curve?

Define Euler's number e as the base for which y=e^x has gradient 1 at the y-intercept, work with the natural logarithm ln x = log_e x as the inverse of e^x, sketch y=e^x and y=ln x as reflections in y=x, transform y=e^x, and solve e^x=k and ln x=k by converting between the two forms

The Year 11 Maths Advanced answer on Euler's number e and natural logarithms: why e (about 2.718) is the base whose graph has gradient exactly 1 at (0,1), the natural log ln x as the inverse of e^x, sketching the reflected pair in y=x, transforming y=e^x, and solving e^x=k and ln x=k, with code-checked numbers and diagrams.

19 min answer →