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Lesson 7 — Logarithmic functions

Logarithm as the inverse of the exponential. Properties. Natural log ln and log base 10.

Used in: Year 1 high school (age 15) · Chemistry (pH) · Engineering (decibel)

\log_a x = y \iff a^y = x

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Rigorous notation, full derivation, hypotheses

Definition and properties

"The logarithmic function with base $b$ , $f(x) = \log_b(x)$ , is the inverse of the exponential function $g(x) = b^x$ . The domain is $(0, +\infty)$ and the range is $\mathbb{R}$ ." — OpenStax Algebra and Trigonometry 2e §6.3

Operational properties

(P1)

what this means · Log of product = sum of logs.

(P2)

what this means · Log of quotient = difference of logs.

(P3)

what this means · Log of power = exponent times log.

(P4)

what this means · Change of base.

"The product rule, quotient rule, and power rule allow you to rewrite logarithmic expressions, transforming products into sums, quoti into differences, and powers into products." — OpenStax Algebra and Trigonometry 2e §6.5

e^x and ln x are reflections of each other across the line y = x. Every inverse function has this property.

Worked examples

Five examples in increasing difficulty — from direct log calculation to pH modeling in chemistry. Each cites an open source.

Example— 2· Example 2 — Use operational properties (intermediate)

Problem: Simplify $\log_2 8 + \log_2 4 - \log_2 16$ .

Strategy: Use the properties $\log_a(xy) = \log_a x + \log_a y$ and $\log_a(x/y) = \log_a x - \log_a y$ to reduce everything to one log.

Solution:

Combine terms via properties: $\log_2 8 + \log_2 4 - \log_2 16 = \log_2 \dfrac{8 \cdot 4}{16} = \log_2 \dfrac{32}{16} = \log_2 2 = 1$ .

Alternative (compute separately): $\log_2 8 = 3$ , $\log_2 4 = 2$ , $\log_2 16 = 4$ . Sum: $3 + 2 - 4 = 1$ . ✓

Verification: $2^1 = 2$ , and the simplified expression gave $\log_2 2 = 1$ . ✓

Source. OpenStax — Algebra and Trigonometry 2e §6.5 — CC-BY 4.0 license.

Example— 3· Example 3 — Logarithmic equation (intermediate)

Problem: Solve $\log_2 x + \log_2 (x - 2) = 3$ .

Strategy: Use the product property $\log_a u + \log_a v = \log_a(uv)$ to combine into one log, then convert to exponential form.

Solution:

Combine logs: $\log_2 x + \log_2 (x - 2) = \log_2 [x(x - 2)] = 3$ .
Convert to exponential: $x(x - 2) = 2^3 = 8$ .
Expand and rearrange: $x^2 - 2x = 8 \Rightarrow x^2 - 2x - 8 = 0$ .
Factor or use quadratic formula: $(x - 4)(x + 2) = 0 \Rightarrow x = 4$ or $x = -2$ .
Check domain: $x > 0$ and $x - 2 > 0$ require $x > 2$ . Only $x = 4$ is valid.

Verification: $\log_2 4 + \log_2 2 = 2 + 1 = 3$ . ✓

Source. OpenStax — College Algebra 2e §6.6 — CC-BY 4.0 license.

Exercise list

10 exercises · 2 with worked solution (25%)

Application 10

Ex. 7.1ApplicationAnswer key
Calculate $\log_2 8$ .
Solve online
Ex. 7.2Application
Calculate $\log_{10} 1$ .
Solve online
Ex. 7.3Application
Calculate $\log_3 9$ .
Solve online
Ex. 7.4Application
Calculate $\log_2(1/4)$ .
Solve online
Ex. 7.5Application
Calculate $\log_5 5$ .
Solve online
Ex. 7.6Application
Calculate $\log_{10} 100$ .
Solve online
Ex. 7.7Application
Calculate $\ln(e^3)$ .
Solve online
Ex. 7.8ApplicationAnswer key
Calculate $\log_4 2$ .
Solve online
Ex. 7.9Application
Calculate $\log_{10} 0.1$ .
Solve online
Ex. 7.10Application
Calculate $\ln 1$ .
Solve online

Sources

Only books that directly fed the text and exercises. Full catalog at /livros.

OpenStax Algebra and Trigonometry 2e — OpenStax · 2022 · EN · CC-BY 4.0 · §6.3–6.7 (definition, properties, equations, models, applications). Primary source.
OpenStax College Algebra 2e — Jay Abramson et al. · 2022, 2nd ed · EN · CC-BY 4.0 · §6.3–6.6 (logarithms and logarithmic equations).
Stitz–Zeager Precalculus — Carl Stitz, Jeff Zeager · 2013, v3 · EN · CC-BY-NC-SA · §6.3 (change of base, equations).