Logarithms
by aoum, Mar 29, 2025, 10:41 PM
Logarithms: Properties, Proofs, and Applications
1. Introduction to Logarithms
A logarithm is the inverse function of exponentiation. It answers the question:
![\[
b^x = y \quad \Rightarrow \quad \log_b y = x.
\]](//latex.artofproblemsolving.com/2/e/b/2ebbf625e50c467c6853ac9f32f3a795999f311a.png)
This definition means that logarithms allow us to determine the exponent required to reach a given number using a specific base.
Example: Since
![\[
2^5 = 32,
\]](//latex.artofproblemsolving.com/3/6/5/3650bd2e0aea69ec9470fca7f073569993ad6a4e.png)
we conclude that
![\[
\log_2 32 = 5.
\]](//latex.artofproblemsolving.com/9/e/a/9ea0ca08420776ffee60a8493f9eae6468a2a2a1.png)
Logarithms are useful for solving exponential equations, simplifying large computations, and modeling growth or decay.
2. Common Logarithms and Special Cases
There are several common logarithmic bases:
Key special cases include:
3. Fundamental Properties of Logarithms
Logarithms obey several important algebraic properties:
4. Solving Logarithmic and Exponential Equations
Example 1: Solve for
in 
.
Since
, we conclude 
. Alternatively, taking 
on both sides:
![\[
x = \log_5 625.
\]](//latex.artofproblemsolving.com/e/c/0/ec052679c5b498f8a0b53e258b8da57cc358ae26.png)
Example 2: Solve
.
Rewriting in exponential form:
![\[
x = 3^4 = 81.
\]](//latex.artofproblemsolving.com/4/2/6/4260a16d961480fd68dcd91322ef71767a1d4888.png)
5. Logarithms in Calculus
Logarithms play a crucial role in differentiation and integration.
Example: Differentiate
.
Using the formula:
![\[
f'(x) = \frac{1}{x \ln 2}.
\]](//latex.artofproblemsolving.com/4/3/6/43630c9b5439af61b1efa6be754613a23b41a393.png)
6. Logarithmic and Exponential Growth
Many natural processes follow logarithmic or exponential growth patterns:
7. Logarithms in Computer Science and Complexity Theory
Logarithms frequently appear in computational complexity:
8. Logarithmic Identities and Approximations
9. Conclusion
Logarithms are a fundamental mathematical tool with applications in algebra, calculus, physics, finance, and computer science. Their properties simplify large calculations, their derivatives and integrals appear in calculus, and their role in complexity theory helps analyze algorithms.
10. Logarithms Video by Sohil Rathi
References
1. Introduction to Logarithms
A logarithm is the inverse function of exponentiation. It answers the question:
![\[
b^x = y \quad \Rightarrow \quad \log_b y = x.
\]](http://latex.artofproblemsolving.com/2/e/b/2ebbf625e50c467c6853ac9f32f3a795999f311a.png)
This definition means that logarithms allow us to determine the exponent required to reach a given number using a specific base.
Plots of logarithm functions, with three commonly used bases. The special points 
are indicated by dotted lines, and all curves intersect in 
.
are indicated by dotted lines, and all curves intersect in 
.
Example: Since
![\[
2^5 = 32,
\]](http://latex.artofproblemsolving.com/3/6/5/3650bd2e0aea69ec9470fca7f073569993ad6a4e.png)
we conclude that
![\[
\log_2 32 = 5.
\]](http://latex.artofproblemsolving.com/9/e/a/9ea0ca08420776ffee60a8493f9eae6468a2a2a1.png)
Logarithms are useful for solving exponential equations, simplifying large computations, and modeling growth or decay.
2. Common Logarithms and Special Cases
There are several common logarithmic bases:
- Common Logarithm:
is shorthand for 
. - Natural Logarithm:
represents 
, where 
. - Binary Logarithm:
appears frequently in computer science.
is shorthand for 
.
represents 
,
.
appears frequently in computer science.Key special cases include:
- since
. - since
. 
since exponentiation and logarithms are inverse operations.
, another demonstration of the inverse relationship.
.
.
since exponentiation and logarithms are inverse operations.
,3. Fundamental Properties of Logarithms
Logarithms obey several important algebraic properties:
- Product Rule:
![\[
\log_b (xy) = \log_b x + \log_b y.
\]](//latex.artofproblemsolving.com/c/d/8/cd8f5f70ee5b81639e44fb35d50371efc8c7f44f.png)
Proof: Let
and 
. Then:
![\[
b^a = x, \quad b^b = y.
\]](//latex.artofproblemsolving.com/5/8/c/58ca896c034225b8ccccfea70608ecf14dbf55d9.png)
Multiplying both sides:
![\[
b^a \cdot b^b = xy.
\]](//latex.artofproblemsolving.com/4/4/8/448656bd0c83afd87cb97c180b38e053f4b493c3.png)
Applying exponent rules:
![\[
b^{a+b} = xy.
\]](//latex.artofproblemsolving.com/0/9/3/09380184cdcb186ad0784e9722d54a49b81c90a9.png)
Taking
on both sides:
![\[
a + b = \log_b xy.
\]](//latex.artofproblemsolving.com/6/0/4/604bb8328209b10e2cf6d19d64086918570cbc08.png)
Thus,
![\[
\log_b x + \log_b y = \log_b (xy).
\]](//latex.artofproblemsolving.com/0/4/9/0491edaebce769911e8e8106b417f53b42e3210b.png)
- Quotient Rule:
![\[
\log_b \left( \frac{x}{y} \right) = \log_b x - \log_b y.
\]](//latex.artofproblemsolving.com/3/7/b/37bf0d5874da5966dcad469c6aafaa50079ae128.png)
This follows from the product rule since
![\[
\log_b x - \log_b y = \log_b \left(\frac{x}{y}\right).
\]](//latex.artofproblemsolving.com/d/9/9/d9908d38ae54a9eeae3d65d7ce06feeff1d8761f.png)
- Power Rule:
![\[
\log_b (x^c) = c \log_b x.
\]](//latex.artofproblemsolving.com/a/3/e/a3e8966e255daec41b0a193c8ae8fb2ced6e2b13.png)
Proof: Let
, so 
. Then:
![\[
b^{cy} = x^c.
\]](//latex.artofproblemsolving.com/e/a/5/ea5e976c85f66e1e31c537a62e1504cab68b8e7e.png)
Taking the logarithm on both sides:
![\[
\log_b x^c = \log_b b^{cy} = cy.
\]](//latex.artofproblemsolving.com/0/8/b/08bd7e20f286aa20e9c3dfa499317ba4b2f5c5cd.png)
Since
, we conclude
![\[
\log_b x^c = c \log_b x.
\]](//latex.artofproblemsolving.com/1/3/b/13be357e1f624366e7f51611bc236ee429c49ee2.png)
- Change of Base Formula:
![\[
\log_b x = \frac{\log_k x}{\log_k b}.
\]](//latex.artofproblemsolving.com/7/2/1/7211e3b1d0a18012610dd6892e63b780fe788e1f.png)
This formula allows logarithms to be converted between different bases. A common case is rewriting any logarithm in terms of base
:
![\[
\log_b x = \frac{\log x}{\log b}.
\]](//latex.artofproblemsolving.com/b/5/6/b56d34d70627044a43d28fab6a85fe7bd22587b6.png)
![\[
\log_b (xy) = \log_b x + \log_b y.
\]](http://latex.artofproblemsolving.com/c/d/8/cd8f5f70ee5b81639e44fb35d50371efc8c7f44f.png)
and 
.![\[
b^a = x, \quad b^b = y.
\]](http://latex.artofproblemsolving.com/5/8/c/58ca896c034225b8ccccfea70608ecf14dbf55d9.png)
![\[
b^a \cdot b^b = xy.
\]](http://latex.artofproblemsolving.com/4/4/8/448656bd0c83afd87cb97c180b38e053f4b493c3.png)
![\[
b^{a+b} = xy.
\]](http://latex.artofproblemsolving.com/0/9/3/09380184cdcb186ad0784e9722d54a49b81c90a9.png)
on both sides:![\[
a + b = \log_b xy.
\]](http://latex.artofproblemsolving.com/6/0/4/604bb8328209b10e2cf6d19d64086918570cbc08.png)
![\[
\log_b x + \log_b y = \log_b (xy).
\]](http://latex.artofproblemsolving.com/0/4/9/0491edaebce769911e8e8106b417f53b42e3210b.png)
![\[
\log_b \left( \frac{x}{y} \right) = \log_b x - \log_b y.
\]](http://latex.artofproblemsolving.com/3/7/b/37bf0d5874da5966dcad469c6aafaa50079ae128.png)
![\[
\log_b x - \log_b y = \log_b \left(\frac{x}{y}\right).
\]](http://latex.artofproblemsolving.com/d/9/9/d9908d38ae54a9eeae3d65d7ce06feeff1d8761f.png)
![\[
\log_b (x^c) = c \log_b x.
\]](http://latex.artofproblemsolving.com/a/3/e/a3e8966e255daec41b0a193c8ae8fb2ced6e2b13.png)
,
.![\[
b^{cy} = x^c.
\]](http://latex.artofproblemsolving.com/e/a/5/ea5e976c85f66e1e31c537a62e1504cab68b8e7e.png)
![\[
\log_b x^c = \log_b b^{cy} = cy.
\]](http://latex.artofproblemsolving.com/0/8/b/08bd7e20f286aa20e9c3dfa499317ba4b2f5c5cd.png)
,![\[
\log_b x^c = c \log_b x.
\]](http://latex.artofproblemsolving.com/1/3/b/13be357e1f624366e7f51611bc236ee429c49ee2.png)
![\[
\log_b x = \frac{\log_k x}{\log_k b}.
\]](http://latex.artofproblemsolving.com/7/2/1/7211e3b1d0a18012610dd6892e63b780fe788e1f.png)
:![\[
\log_b x = \frac{\log x}{\log b}.
\]](http://latex.artofproblemsolving.com/b/5/6/b56d34d70627044a43d28fab6a85fe7bd22587b6.png)
4. Solving Logarithmic and Exponential Equations
- To solve
, apply logarithms:
![\[
x = \log_b y.
\]](//latex.artofproblemsolving.com/f/0/7/f07e6e5096077d75239cc4a78024c5cfeb12738b.png)
- To solve logarithmic equations, use exponentiation: If
![\[
\log_b x = y,
\]](//latex.artofproblemsolving.com/d/5/f/d5f313e3e218eae8e7fa8455f796ff87bd9cfcf1.png)
then
![\[
x = b^y.
\]](//latex.artofproblemsolving.com/d/1/7/d17d3455324cf7c4f2bb40572d04ef0d9d671bec.png)
,![\[
x = \log_b y.
\]](http://latex.artofproblemsolving.com/f/0/7/f07e6e5096077d75239cc4a78024c5cfeb12738b.png)
![\[
\log_b x = y,
\]](http://latex.artofproblemsolving.com/d/5/f/d5f313e3e218eae8e7fa8455f796ff87bd9cfcf1.png)
![\[
x = b^y.
\]](http://latex.artofproblemsolving.com/d/1/7/d17d3455324cf7c4f2bb40572d04ef0d9d671bec.png)
Example 1: Solve for
in 
.Since
, we conclude 
. Alternatively, taking 
on both sides:![\[
x = \log_5 625.
\]](http://latex.artofproblemsolving.com/e/c/0/ec052679c5b498f8a0b53e258b8da57cc358ae26.png)
Example 2: Solve
.Rewriting in exponential form:
![\[
x = 3^4 = 81.
\]](http://latex.artofproblemsolving.com/4/2/6/4260a16d961480fd68dcd91322ef71767a1d4888.png)
5. Logarithms in Calculus
Logarithms play a crucial role in differentiation and integration.
- The derivative of the natural logarithm:
![\[
\frac{d}{dx} \ln x = \frac{1}{x}, \quad x > 0.
\]](//latex.artofproblemsolving.com/f/0/3/f0350fe88c97d98d71c3d89d9bfff1d2b184ba39.png)
- The derivative of a logarithm with arbitrary base
:
![\[
\frac{d}{dx} \log_b x = \frac{1}{x \ln b}.
\]](//latex.artofproblemsolving.com/b/c/3/bc3c57baaab411da48bef36da123dce267f40b57.png)
- The integral of the natural logarithm:
![\[
\int \ln x \,dx = x \ln x - x + C.
\]](//latex.artofproblemsolving.com/5/8/c/58c695831d8f444d14442845860b626f214b4648.png)
![\[
\frac{d}{dx} \ln x = \frac{1}{x}, \quad x > 0.
\]](http://latex.artofproblemsolving.com/f/0/3/f0350fe88c97d98d71c3d89d9bfff1d2b184ba39.png)
:![\[
\frac{d}{dx} \log_b x = \frac{1}{x \ln b}.
\]](http://latex.artofproblemsolving.com/b/c/3/bc3c57baaab411da48bef36da123dce267f40b57.png)
![\[
\int \ln x \,dx = x \ln x - x + C.
\]](http://latex.artofproblemsolving.com/5/8/c/58c695831d8f444d14442845860b626f214b4648.png)
Example: Differentiate
.Using the formula:
![\[
f'(x) = \frac{1}{x \ln 2}.
\]](http://latex.artofproblemsolving.com/4/3/6/43630c9b5439af61b1efa6be754613a23b41a393.png)
6. Logarithmic and Exponential Growth
Many natural processes follow logarithmic or exponential growth patterns:
- Population growth follows the exponential model
![\[
P(t) = P_0 e^{rt}.
\]](//latex.artofproblemsolving.com/b/f/c/bfc994731c65e69c13c3f0b532fcbfdfa0c93741.png)
- The time required for a radioactive substance to decay is given by
![\[
t = \frac{\ln (A_0 / A)}{\lambda}.
\]](//latex.artofproblemsolving.com/a/1/b/a1b61b405b580cb24e8fc71977252e9079a74c25.png)
- The pH scale in chemistry is logarithmic:
![\[
\text{pH} = -\log_{10} [\text{H}^+].
\]](//latex.artofproblemsolving.com/2/5/8/25825d5f625a4db07a408e048d06d0707b1632ee.png)
![\[
P(t) = P_0 e^{rt}.
\]](http://latex.artofproblemsolving.com/b/f/c/bfc994731c65e69c13c3f0b532fcbfdfa0c93741.png)
![\[
t = \frac{\ln (A_0 / A)}{\lambda}.
\]](http://latex.artofproblemsolving.com/a/1/b/a1b61b405b580cb24e8fc71977252e9079a74c25.png)
![\[
\text{pH} = -\log_{10} [\text{H}^+].
\]](http://latex.artofproblemsolving.com/2/5/8/25825d5f625a4db07a408e048d06d0707b1632ee.png)
7. Logarithms in Computer Science and Complexity Theory
Logarithms frequently appear in computational complexity:
- The runtime of binary search is
. - The number of bits required to represent an integer
in binary is approximately 
. - The time complexity of many divide-and-conquer algorithms, such as merge sort, is
.
.
in binary is approximately 
.
.8. Logarithmic Identities and Approximations
- Stirling’s Approximation:
![\[
\ln n! \approx n \ln n - n.
\]](//latex.artofproblemsolving.com/4/9/c/49c7d8d753413fc845a44b9742a9b20b3fc15e24.png)
- Napier’s Logarithms: Early logarithms introduced by John Napier were based on geometric sequences.
- Approximations: For small ,
![\[
\ln(1 + x) \approx x \quad \text{for } x \approx 0.
\]](//latex.artofproblemsolving.com/d/d/0/dd0c4cb4c3b860eb065406a64e1141cebd68324b.png)
- The Arithmetic-Geometric Mean Approximation:
![\[
\log_b x \approx \frac{x - 1}{x + 1}.
\]](//latex.artofproblemsolving.com/0/d/d/0ddb5a3d0344539f6e84ae009227e092d0272ca0.png)
![\[
\ln n! \approx n \ln n - n.
\]](http://latex.artofproblemsolving.com/4/9/c/49c7d8d753413fc845a44b9742a9b20b3fc15e24.png)
![\[
\ln(1 + x) \approx x \quad \text{for } x \approx 0.
\]](http://latex.artofproblemsolving.com/d/d/0/dd0c4cb4c3b860eb065406a64e1141cebd68324b.png)
![\[
\log_b x \approx \frac{x - 1}{x + 1}.
\]](http://latex.artofproblemsolving.com/0/d/d/0ddb5a3d0344539f6e84ae009227e092d0272ca0.png)
9. Conclusion
Logarithms are a fundamental mathematical tool with applications in algebra, calculus, physics, finance, and computer science. Their properties simplify large calculations, their derivatives and integrals appear in calculus, and their role in complexity theory helps analyze algorithms.
10. Logarithms Video by Sohil Rathi
References
April 2025
March 2025