Logarithms: Equations, Systems and Properties

• Introduction

• Remember that... definition of logarithm and its properties

• Resolved Logarithmic Equations, Systems and Demonstration of the properties

Introduction

A logarithmic equation is an equation that has an unknown factor in the argument of a logarithm. In reality, the resolution is reduced to the resolution of equations of the same type as the expressions in the arguments (quadratic equations, cubic equations, irrational equations...).

This section is a collection of resolved exercises of equations and equation systems of this kind, which are intended to be in order of increasing difficulty.

In the majority of logarithms the base is not specified, because we suppose it's 10. Although, in this sense, we must decide that in the majority of scientific texts, if it does not say otherwise, that the base is e (as it's known, Napierian logarithm).

Also, at the end we will prove the logarithmic properties: logarithm of a product, of a quotient, of a power and base switch.

Regarding the use of the logarithms, we can talk of their frequent use in physics. They appear, for example, to calculate the apparent magnitude of a celestial body (measured by the amount of light we perceive); the magnitude of an earthquake in the Richter scale; the age (time gone by) in radiometric dating (the Carbon 14 test); the pH.

Remember that...

Before starting the exercises, let's remember the definition of logarithm:

$$log_b (a) = c \Leftrightarrow b^c = a$$

• a is called the base of the logarithm

• b is called the argument of the logarithm

• c the number that

$$b^c = a$$

• So, c is the exponent to which the base b must be raised to be the number a.

$$b^{log_b (c)} = c$$

• If b is the number e, we write ln(x) instead loge (x). This is the Naperian logarithm.

Logarithm Properties

 logarithm of a product: logarithm of a quotient: logarithm of a power: change of base: Useful property:

Exercise 1

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Equation 1

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Equation 2

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Equation 3

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Equation 4

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Equation 5

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Equation 6

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Equation 7

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Equation 8

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Equation 9

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Equation 10

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Equation 11

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Equation 12

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Equation 13

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Equation 14

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Equation 15

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Equation 16

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Equation 17

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Equation 18

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Equation 19

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Equation 20

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Equation 21

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Equation 22

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Equation 23

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Equation 24

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Equation 25

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System 1

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System 2

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System 3

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System 4

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System 5

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System 6

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System 7

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System 8

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System 9

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System 10

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System 11

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Proof of the Properties

Property 1: logarithm of a product

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Property 2: logarithm of a quotient

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Property 3: logarithm of a power

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Property 4: change of base

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Property 5: change of base

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