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Longhand

Before the midterm

Solving for an exponent without a logarithm

The habit: Set the exponential equal to the target ratio and take logarithms of both sides to bring the unknown down.

3 practice questions are built to catch it, in 3 topics.

Practise against it: 2 questions

What it looks like

A question from functions, graphs and inverses, with the answer this slip gives beside the right one.

Standard2 marks · 4 min

Let f(x)=3e2x−1f(x) = 3e^{2x} - 1. Find f−1(5)f^{-1}(5).

With the slip
1\displaystyle 1
Worked through
f−1(5)=log⁡22≈0.347\displaystyle f^{-1}(5) = \dfrac{\log 2}{2} \approx 0.347

In e2x = 2, the right side is 2 = elog 2, so 2x = log 2, not 2x = 2.

See the worked solution

Where it comes up

Practise against it

Each of these has an answer this slip gives. Work them on paper: if you fall into it, the verdict names it and the question comes back in review.

  1. 01
    A bacterial culture has population P(t)=5e0.2tP(t) = 5e^{0.2t} million after tt hours. How long does it take the population to double?

    Exponential growth and decayIntroductory

  2. 02
    Which list gives every horizontal and vertical asymptote of the graph of the function below? f(x)=3ex+2ex−1\displaystyle f(x) = \frac{3e^{x} + 2}{e^{x} - 1}

    LimitsStandard

Start the set