MathLab with Mr. Ahmed

MathLab with Mr. Ahmed is a math learning website by Ahmed Amasha, a mathematics teacher teaching the American curriculum. Explore Algebra 2, Precalculus, Calculus, and SAT and PSAT Math Prep through lessons, focused practice and printable resources.

MathLabwith Mr. Ahmed
Logarithmic, Exponential, and Other Transcendental Functions

5.5 Bases Other than e and Applications

Focus on other exponential and logarithmic bases, doubling time, exponential models from data.

Free preview Explore the lesson before creating an account.

Play games for this section

Before you solve

Ideas in this lesson

01Other exponential and logarithmic basesWorked steps · five questions02Doubling timeWorked steps · five questions03Exponential models from dataWorked steps · five questions
Idea 1 of 3

Other exponential and logarithmic bases

Differentiating bxb^{x} introduces ln⁡b\ln b. Differentiating log⁡bx\log _b x introduces division by ln⁡b\ln b.

Worked example 1

Find the derivative at x=0x = 0 of 2x2^{x}.
  1. f′(x)=2xln⁡2f'(x) = 2^{x} \ln 2
  2. f′(0)=ln⁡2≈0.69f'(0) = \ln 2 \approx 0.69

Answer: ln⁡2≈0.69\ln 2 \approx 0.69

Worked example 2

Find the derivative at x=1x = 1 of log⁡3x\log _{3} x.
  1. f′(x)=1xln⁡3f'(x) = \dfrac{1}{x \ln 3}
  2. f′(1)=1ln⁡3≈0.91f'(1) = \dfrac{1}{\ln 3} \approx 0.91

Answer: 1ln⁡3≈0.91\dfrac{1}{\ln 3} \approx 0.91

Idea 2 of 3

Doubling time

For Q=Q0ektQ = Q_{0}e^{kt} with k>0k > 0, doubling time is ln⁡2k\frac{\ln 2}{k}. It does not depend on Q0Q_{0}.

Worked example 1

A quantity follows Q(t)=2e0.1tQ(t) = 2e^{0.1t}. Find Q(2)Q(2).
  1. Q(t)=2e0.1tQ(t) = 2e^{0.1t}; Q′(t)=0.1Q(t)Q'(t) = 0.1Q(t)
  2. Q(2)=2e0.2=2e0.2≈2.44Q(2) = 2e^{0.2} = 2e^{0.2} \approx 2.44

Answer: 2e0.2≈2.442e^{0.2} \approx 2.44

Worked example 2

A quantity follows Q(t)=3e0.2tQ(t) = 3e^{0.2t}. Find its doubling time.
  1. Q(t)=3e0.2tQ(t) = 3e^{0.2t}; Q′(t)=0.2Q(t)Q'(t) = 0.2Q(t)
  2. Doubling: e0.2t=2  ⟹  t=ln⁡20.2=ln⁡20.2≈3.47e^{0.2t} = 2 \implies t = \dfrac{\ln 2}{0.2} = \dfrac{\ln 2}{0.2} \approx 3.47

Answer: ln⁡20.2≈3.47\dfrac{\ln 2}{0.2} \approx 3.47

Idea 3 of 3

Exponential models from data

Equal multiplicative changes suggest an exponential model. Distinguish discrete factors from continuous relative rates.

Worked example 1

A population starts at 22 and doubles each hour. Find P(1)P(1).
  1. P(t)=2⋅2tP(t) = 2 \cdot 2^{t}
  2. P(1)=2⋅2=4P(1) = 2 \cdot 2 = 4

Answer: 44

Worked example 2

A population starts at 33 and doubles each hour. At what time does the population reach 1212?
  1. P(t)=3⋅2tP(t) = 3 \cdot 2^{t}
  2. 2t=4  ⟹  t=22^{t} = 4 \implies t = 2

Answer: 22

Resource alignment: Larson & Edwards, Calculus of a Single Variable, Tenth Edition, §5.5,p5.5, p. 356356. Explanations, diagrams and numerical practice variants are original MathLab content aligned to this section.