MYP Extended Mathematics Topic Logarithms Revision Notes

MYP Extended Mathematics Topic Logarithms Revision Notes | RevisionDojo

Logarithms as Inverse Exponentials

Definition

Logarithm

For $a>0$, $a\neq 1$, and $x>0$, $\log_a x$ is the exponent $k$ such that $a^k=x$.

This definition immediately gives the key restrictions:
1. The base must satisfy $a>0$ and $a\neq 1$.
2. The argument (the input to the log) must satisfy $x>0$.
Two bases are especially common:
1. Common logarithm: $\log x$ means $\log_{10} x$.
2. Natural logarithm: $\ln x$ means $\log_e x$, where $e\approx 2.71828$.

Note

Historically, logarithms were developed to simplify computations involving very large numbers (John Napier, 1614).
Modern calculators compute logs directly, but the algebraic rules remain central for solving problems.

Converting between Logarithmic and Exponential Form

If you can switch forms fluently, most log questions become much easier.
$\log_a x = y$ means $a^y = x$.
$a^y = x$ means $\log_a x = y$.

Example

Rewrite $\log_2 8 = 3$ in exponential form.
Since "log equals exponent", $2^3=8$.

Logarithm Laws and Why They Work

The log laws mirror the exponent laws because logarithms are defined through exponentials.

Product Rule

Definition

Product rule

$\log_a(xy)=\log_a x+\log_a y$.

Proof sketch:

Let $p=\log_a x$ and $q=\log_a y$. Then $a^p=x$ and $a^q=y$.
Multiply: $$a^p\cdot a^q=a^{p+q}=xy$$

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Logarithms as Inverse Exponentials

Definition

Logarithm

For $a>0$, $a\neq 1$, and $x>0$, $\log_a x$ is the exponent $k$ such that $a^k=x$.

This definition immediately gives the key restrictions:
1. The base must satisfy $a>0$ and $a\neq 1$.
2. The argument (the input to the log) must satisfy $x>0$.
Two bases are especially common:
1. Common logarithm: $\log x$ means $\log_{10} x$.
2. Natural logarithm: $\ln x$ means $\log_e x$, where $e\approx 2.71828$.

Note

Historically, logarithms were developed to simplify computations involving very large numbers (John Napier, 1614).
Modern calculators compute logs directly, but the algebraic rules remain central for solving problems.

Converting between Logarithmic and Exponential Form

If you can switch forms fluently, most log questions become much easier.
$\log_a x = y$ means $a^y = x$.
$a^y = x$ means $\log_a x = y$.

Example

Rewrite $\log_2 8 = 3$ in exponential form.
Since "log equals exponent", $2^3=8$.

Logarithm Laws and Why They Work

The log laws mirror the exponent laws because logarithms are defined through exponentials.

Product Rule

Definition

Product rule

$\log_a(xy)=\log_a x+\log_a y$.

Proof sketch:

Let $p=\log_a x$ and $q=\log_a y$. Then $a^p=x$ and $a^q=y$.
Multiply: $$a^p\cdot a^q=a^{p+q}=xy$$

Definition

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(on a website) an arrangement whereby access is restricted to users who have paid to subscribe to the site.

anim nostrud sit dolore minim proident quis fugiat velit et eiusmod nulla quis nulla mollit dolor sunt culpa aliqua

Duis aute irure dolor in reprehenderit

Note

Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam quis nostrud exercitation.

Excepteur sint occaecat cupidatat non proident

Tip

Lorem ipsum dolor sit amet, consectetur adipiscing elit.
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Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris.
Duis aute irure dolor in reprehenderit in voluptate velit esse cillum.

Advanced Number and Bounds3 subtopics

Inequalities and Modelling1 subtopic

Function Transformations4 subtopics

Advanced Probability1 subtopic

Geometric Transformations1 subtopic

Variation and Standard Deviation3 subtopics

Trigonometric Functions and Rules4 subtopics

Sequences, Rational Expressions and Equations5 subtopics

Covariance and Correlation3 subtopics

Logarithms and Non-Linear Inequalities3 subtopics

Graph Theory and Vectors4 subtopics

MYP Extended Mathematics Logarithms Notes

Advanced Number and Bounds3 subtopics

Inequalities and Modelling1 subtopic

Function Transformations4 subtopics

Advanced Probability1 subtopic

Geometric Transformations1 subtopic

Variation and Standard Deviation3 subtopics

Trigonometric Functions and Rules4 subtopics

Sequences, Rational Expressions and Equations5 subtopics

Covariance and Correlation3 subtopics

Logarithms and Non-Linear Inequalities3 subtopics

Graph Theory and Vectors4 subtopics

Logarithms as Inverse Exponentials

Converting between Logarithmic and Exponential Form

Logarithm Laws and Why They Work

Product Rule

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Logarithms as Inverse Exponentials

Converting between Logarithmic and Exponential Form

Logarithm Laws and Why They Work

Product Rule

anim nostrud sit dolore minim proident quis fugiat velit et eiusmod nulla quis nulla mollit dolor sunt culpa aliqua

Duis aute irure dolor in reprehenderit

Excepteur sint occaecat cupidatat non proident

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