## What does $2^x$ mean and what is its inverse?

### Key questions

1. 1

What are exponential growth and decay?

2. 2

How are the exponential and logarithmic functions related?

3. 3

How are logarithms used?

4. 4

How can we rewrite expressions involving logarithms?

5. 5

What sorts of equations can we solve using logarithms?

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#### Introducing...

Resource type Title
Building blocks Reach for the stars
Building blocks See the power
Investigation How fast does it grow?

#### Developing...

Resource type Title
Rich example Plotting the planets
Building blocks 1950s calculators
Building blocks Summing to one
Fluency exercise Factorial fragments
Fluency exercise Logarithm lattice
Fluency exercise Logarithm lineup
Package of problems Changing bases
Scaffolded task Proving the laws of logarithms
Problem requiring decisions To log or not to log?
Food for thought How far did the earth move?
Go and think about it... An irrational inequality
Bigger picture Re-discovering the logarithm—Mercator
Resource in action Changing bases - teacher support
Resource in action To log or not to log - teacher support

#### Review questions

Title Ref
Can we evaluate these log expressions without using a calculator? R9976
Can we find $a$ and $b$ as powers of $10$? R9366
Can we find bounds for $\log_2 3$? R9381
Can we find these log expressions in terms of $\log_9 x$? R7298
Can we solve $\log_3 x + 2 \log_x 3 = -3$? R7845
Can we solve the equation $\log_2 x + \log_3 x = 1 + \log_4 x$? R8260
Can we solve these simultaneous exponential equations? R5940
Can we solve these three simultaneous log equations? R5462
Can we use $\log_{10}$ to find the first digit of $3^{100}$? R9747
Can we use the laws of logs to find $x$ and $y$? R7471
Given this log equation, when can $a$ be largest? R7015
How large is $2^{100}$, and what is its starting digit? R6045
How many real roots does $8^x+4=4^x+2^{x+2}$ have? R7606
How many roots are there to $x=8^{\log_2x} -9^{\log_3x}-4^{\log_2x}+\log_{0.5}{0.25}$? R5103
If $yz=a^2$, can we prove that $1/(a+y) + 1/(a+z) = 1/a$? R6806
If we arrange $2^x, 2x$ and $x^2$ by size, what orders are possible? R5156
When does $9^x - 3^{x+1} = k$ have one or more real solutions? R7384
When does this exponential function equal this linear one? R8013
When does this inequality have a finite number of integer solutions? R6414
Which is the largest of these four logarithms? R7231
Which is the smallest of these surd, log and trig expressions? R5080
Which of these four log expressions is the smallest? R9680
Which of these log and trig expressions is the largest? R7184