MYP Extended Mathematics Topic Logarithmic Equations Revision Notes

MYP Extended Mathematics Topic Logarithmic Equations Revision Notes | RevisionDojo

Exponent and Logarithm Laws Drive Most Solutions

Logarithmic equations are usually solved using one (or both) of these strategies:
1. Rewrite exponentials and logs into an equivalent form.
2. Simplify using exponent and logarithm laws, then solve.
The most used exponent laws are:
1. $a^m\cdot a^n=a^{m+n}$
2. $\dfrac{a^m}{a^n}=a^{m-n}$
3. $(a^m)^n=a^{mn}$
4. $a^{-n}=\dfrac{1}{a^n}$
5. $a^{1/n}=\sqrt[n]{a}$
The matching logarithm laws (for $M>0, N>0$) are:
1. $\log_a(MN)=\log_a M+\log_a N$
2. $\log_a\left(\dfrac{M}{N}\right)=\log_a M-\log_a N$
3. $\log_a(M^k)=k\log_a M$

Note

The logarithm laws are consequences of exponent laws.
For example, $MN=a^p\cdot a^q=a^{p+q}$ leads to $\log_a(MN)=p+q=\log_a M+\log_a N$.

Solve Without a Calculator by Recognizing Powers

If you can express the number as an exact power of the base, you can solve instantly.

Example

$\log_5 625 = x$. Since $625=5^4$, you get $x=4$.
$\log_4 0.0625 = x$. Since $0.0625=\dfrac{1}{16}=4^{-2}$, you get $x=-2$.
$\log_2 0.125 = x$. Since $0.125=\dfrac{1}{8}=2^{-3}$, you get $x=-3$.
$\log_3 1=x$. Since $1=3^0$, you get $x=0$.

Examples with Fractional Exponents

Fractional exponents usually come from roots.

Example

$\log_5 \sqrt{5}=x$:
- Since $\sqrt{5}=5^{1/2}$, you get $x=\dfrac{1}{2}$.
$\log_3\left(3\sqrt{3}\right)=x$:
- Rewrite: $3\sqrt{3}=3\cdot 3^{1/2}=3^{3/2}$, so $x=\dfrac{3}{2}$.

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Exponent and Logarithm Laws Drive Most Solutions

Logarithmic equations are usually solved using one (or both) of these strategies:
1. Rewrite exponentials and logs into an equivalent form.
2. Simplify using exponent and logarithm laws, then solve.
The most used exponent laws are:
1. $a^m\cdot a^n=a^{m+n}$
2. $\dfrac{a^m}{a^n}=a^{m-n}$
3. $(a^m)^n=a^{mn}$
4. $a^{-n}=\dfrac{1}{a^n}$
5. $a^{1/n}=\sqrt[n]{a}$
The matching logarithm laws (for $M>0, N>0$) are:
1. $\log_a(MN)=\log_a M+\log_a N$
2. $\log_a\left(\dfrac{M}{N}\right)=\log_a M-\log_a N$
3. $\log_a(M^k)=k\log_a M$

Note

The logarithm laws are consequences of exponent laws.
For example, $MN=a^p\cdot a^q=a^{p+q}$ leads to $\log_a(MN)=p+q=\log_a M+\log_a N$.

Solve Without a Calculator by Recognizing Powers

If you can express the number as an exact power of the base, you can solve instantly.

Example

$\log_5 625 = x$. Since $625=5^4$, you get $x=4$.
$\log_4 0.0625 = x$. Since $0.0625=\dfrac{1}{16}=4^{-2}$, you get $x=-2$.
$\log_2 0.125 = x$. Since $0.125=\dfrac{1}{8}=2^{-3}$, you get $x=-3$.
$\log_3 1=x$. Since $1=3^0$, you get $x=0$.

Examples with Fractional Exponents

Fractional exponents usually come from roots.

Example

$\log_5 \sqrt{5}=x$:
- Since $\sqrt{5}=5^{1/2}$, you get $x=\dfrac{1}{2}$.
$\log_3\left(3\sqrt{3}\right)=x$:
- Rewrite: $3\sqrt{3}=3\cdot 3^{1/2}=3^{3/2}$, so $x=\dfrac{3}{2}$.

Definition

Paywall

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

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Duis aute irure dolor in reprehenderit

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Tip

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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 Logarithmic Equations 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

Exponent and Logarithm Laws Drive Most Solutions

Solve Without a Calculator by Recognizing Powers

Examples with Fractional Exponents

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Duis aute irure dolor in reprehenderit

Excepteur sint occaecat cupidatat non proident

Unlock the rest of this chapter with a free account

Exponent and Logarithm Laws Drive Most Solutions

Solve Without a Calculator by Recognizing Powers

Examples with Fractional Exponents

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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