3.2 Logarithmic and exponential functions

4 of 4 objectives have lessons · 0 studied · 0 secure

Lessons

Logarithms are indices in disguise$\log_a x$ answers "what power of $a$ gives $x$?" — so every law of logarithms is a law of indices rewritten. 7 min, not started
e^x and ln x: a pair of inverses$e \approx 2.718$ is the natural base; $e^x$ and $\ln x$ undo each other, their graphs are mirror images in $y = x$, and $\ln$ only accepts positive inputs. 8 min, not started
Solving equations and inequalities with x in the indexWhen the unknown is an exponent, take logarithms of both sides — the power law brings the exponent down where you can solve for it. 8 min, not started
Turning curves into straight linesTaking logs converts $y = kx^n$ and $y = ka^x$ into straight-line relationships, so the gradient and intercept reveal the unknown constants. 8 min, not started

4 objectives

What Cambridge says you should be able to do, in our words. Codes are the topic number plus the objective’s position.

3.2.1

Understand the relationship between logarithms and indices, and use the laws of logarithms (excluding change of base).

Taught in: Logarithms are indices in disguise · 7 questions

3.2.2

Understand the definition and properties of eˣ and ln x, including their relationship as inverse functions and their graphs.

Includes knowing the graph of y = e^(kx) for both positive and negative values of k.

Taught in: e^x and ln x: a pair of inverses · 7 questions

3.2.3

Use logarithms to solve equations and inequalities in which the unknown appears in indices.

Examples: 2ˣ < 5, 3 × 2^(3x − 1) < 5, 3^(x + 1) = 4^(2x − 1).

Taught in: Solving equations and inequalities with x in the index · 8 questions

3.2.4

Use logarithms to transform a given relationship to linear form, and hence determine unknown constants by considering the gradient and/or intercept.

Examples: y = kxⁿ gives ln y = ln k + n ln x, which is linear in ln x and ln y; y = k(aˣ) gives ln y = ln k + x ln a, which is linear in x and ln y.

Taught in: Turning curves into straight lines · 7 questions