Logarithm

Logarithms: In-Depth Explanation of Course Arithmetic Aptitude

Logarithms are a fundamental concept in mathematics, especially in fields involving quantitative analysis like business administration. Understanding logarithms helps in solving exponential equations, analyzing growth patterns, and much more. This section covers logarithms in detail, exploring their properties, applications, and related subtopics.

What is a Logarithm?

A logarithm is the inverse operation of exponentiation. It tells us how many times one number, called the base, must be multiplied by itself to get another number. For a given number $x$ and a base $b$, the logarithm of $x$ to the base $b$ is written as ${log}_b\left(x\right)$. This means:

${log}_b\left(x\right)=y$ if and only if $b^y=x$

For example, ${log}_2\left(8\right)=3$ because $2^3=8$.

Basic Properties of Logarithms

Product Rule

The logarithm of a product is the sum of the logarithms of the factors.

${log}_b(M\cdot N)={log}_b\left(M\right)+{log}_b\left(N\right)$

Example:

${log}_2(16\cdot 4)={log}_2\left(16\right)+{log}_2\left(4\right)=4+2=6$

Quotient Rule

The logarithm of a quotient is the difference of the logarithms.

${log}_b\left(\frac{M}{N}\right)={log}_b\left(M\right)-{log}_b\left(N\right)$

Example:

${log}_2\left(\frac{16}{4}\right)={log}_2\left(16\right)-{log}_2\left(4\right)=4-2=2$

Power Rule

The logarithm of a number raised to a power is the power times the logarithm of the number.

${log}_b\left(M^k\right)=k\cdot {log}_b\left(M\right)$

Example:

${log}_2\left(8^3\right)=3\cdot {log}_2\left(8\right)=3\cdot 3=9$

Change of Base Formula

This formula allows the conversion of logarithms from one base to another.

${log}_b\left(M\right)=\frac{{log}_k}{\left(}$

Example:

${log}_2\left(8\right)=\frac{{log}_{10}}{\left(}\approx \frac{0.903}{0.301}\approx 3$

Logarithm of 1

The logarithm of 1 to any base is 0.

${log}_b\left(1\right)=0$

Example:

${log}_5\left(1\right)=0$

Logarithm of the Base

The logarithm of a base to itself is always 1.

${log}_b\left(b\right)=1$

Example:

${log}_7\left(7\right)=1$

Common Logarithms and Natural Logarithms

Common Logarithms

These are logarithms with base 10, written as ${log}_{10}\left(x\right)$ or simply $log\left(x\right)$.

Example:

$log\left(1000\right)=3$ because ${10}^3=1000$.

Natural Logarithms

These are logarithms with base $e$ (Euler's number, approximately 2.71828), written as $ln\left(x\right)$.

Example:

$ln\left(e^2\right)=2$ because $e^2=e^2$.

Applications of Logarithms

Solving Exponential Equations

Logarithms are used to solve equations where the unknown is an exponent.

Example:

Solve $2^x=16$.

${log}_2\left(2^x\right)={log}_2\left(16\right)$

$x={log}_2\left(16\right)=4$

Compound Interest

Logarithms help in solving compound interest problems, where the amount grows exponentially over time.

Formula: $A=P{\left(1+\frac{r}{n}\right)}^{nt}$

To find $t$:

$A=P{\left(1+\frac{r}{n}\right)}^{nt}$

$log\left(\frac{A}{P}\right)=ntlog(1+\frac{r}{n})$

$t=\frac{\log \frac{A}{P}}{nlog(1+\frac{r}{n})}$

Growth and Decay

Logarithms are used in models of population growth, radioactive decay, and other natural processes.

Example: Population growth $P\left(t\right)=P_0{e}^r$.

To find $t$ when $P\left(t\right)$ is known:

$P\left(t\right)=P_0{e}^r$

$\log \left(P\left(t\right)\right)=log\left(P_{0}ert\right)$

$\log \left(P\left(t\right)\right)=log\left(P_0\right)+rt$

$t=\frac{\log \left(P\right(t\left)\right)-\log \left(p_0\right)}{r}$

pH Calculation in Chemistry

The pH of a solution is determined using logarithms:

$$pH=-log10(H+)

Subtopics in Logarithms

Introduction to Logarithms

• Understanding the basic definition and notation of logarithms.
• Converting between exponential and logarithmic forms.
• Example: ${10}^3=1000$ can be written as ${log}_{10}\left(1000\right)=3$.

Properties and Laws of Logarithms

• Mastering the product, quotient, and power rules.
• Utilizing the change of base formula.
• Example: ${log}_2\left(32\right)=\frac{{log}_{10}}{\left(}\approx \frac{1.505}{0.301}\approx 5$

Graphing Logarithmic Functions

• Understanding the shape and characteristics of logarithmic graphs.
• Exploring transformations such as shifts, reflections, and stretches.
• Example: The graph of $y={log}_b\left(x\right)$ passes through (1,0) and has a vertical asymptote at $x=0$.

Solving Logarithmic Equations

• Techniques for solving equations involving logarithms.
• Applying logarithmic properties to simplify and solve.
• Example: Solve ${log}_3\left(x\right)+{log}_3(x-2)=1$

  $$log3(x(x-2))=1

  $$x(x-2)=3

  $$x2-2x-3=0

  $$(x-3)(x+1)=0

  $x=3$ or $x=-1$

  Only $x=3$ is valid because ${log}_3(-1)$ is undefined.

Applications of Logarithms

• Practical uses in various fields such as finance, science, and engineering.
• Problem-solving involving real-world scenarios.
• Example: In finance, logarithms can be used to calculate the time required for an investment to double at a certain interest rate.

Logarithmic and Exponential Relationships

• Exploring the inverse relationship between logarithmic and exponential functions.
• Solving systems of equations involving both types of functions.
• Example: If $y=b^x$, then $x={log}_b\left(y\right)$.

Summary

• Logarithms are essential tools for solving exponential equations and have broad applications in various fields.
• Key properties include product, quotient, and power rules, and the change of base formula.
• Common logarithms (base 10) and natural logarithms (base $e$) are frequently used in practical problems.
• Applications range from compound interest and growth/decay models to pH calculations and financial analysis.