[b]1. [/b][b]Logarithm of a product[br][br][/b][img]https://www.matesfacil.com/ESO/logaritmos/dlog1.png[/img][br][br][b]2. [/b][b]Logarithm of a quotient[br][br][img]https://www.matesfacil.com/ESO/logaritmos/dlog2.png[/img][br][br]3. [b] Logarithm of a power[br][br][img]https://www.matesfacil.com/ESO/logaritmos/dlog3.png[/img][br][br][/b]4. [b]Change of base[/b] [br][br][img]https://www.matesfacil.com/ESO/logaritmos/dlog4.png[/img][br][br]5. Inverse[br][br][img]https://www.matesfacil.com/ESO/logaritmos/dlog5.png[/img][/b]
A [b]logarithmic equation[/b] is an equation that has an unknown factor in the [b]argument of a logarithm[/b]. 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...).
[b]Example 1[br][br][/b][b][img]https://www.matesfacil.com/ESO/logaritmos/elog1.png[/img][br][br][/b]We will use the properties of logarithms and that [math]log\left(1000\right)=log\left(10^3\right)=3.[/math][br][br][img]https://www.matesfacil.com/ESO/logaritmos/elog1-01.png[/img][br][br]We have an equality between logarithms, so the arguments (what is inside) has to be the same:[br][br][img]https://www.matesfacil.com/ESO/logaritmos/elog1-02.png[/img][br][br]The solution to the equation is [i]x = 50[/i].
[b]Example 2[br][br][/b][img]https://www.matesfacil.com/ESO/logaritmos/elog6.png[/img][br][br][img]https://www.matesfacil.com/ESO/logaritmos/elog6-1.png[/img]
[b]Example 1[br][/b] [br][img]https://www.matesfacil.com/ESO/logaritmos/slog1.png[/img][br][br]We apply the change of variable[br][br][img]https://www.matesfacil.com/ESO/logaritmos/slog1-01.png[/img][br][br]This way we obtain the following linear equation system[br][br][img]https://www.matesfacil.com/ESO/logaritmos/slog1-02.png[/img][br][br]we resolve it and we undo the cange of variable:[br][br][img]https://www.matesfacil.com/ESO/logaritmos/slog1-03.png[/img]
[b]Example 2[br][br][/b][b][img]https://www.matesfacil.com/ESO/logaritmos/slog9.png[/img][br][br][/b]First we apply the property [i]logarithm of a product[/i]:[br][br][img]https://www.matesfacil.com/ESO/logaritmos/slog9-01.png[/img][br][br]We apply the change of variable:[br][br][img]https://www.matesfacil.com/ESO/logaritmos/slog9-02.png[/img][br][br]and we obtain the following system:[br][br][img]https://www.matesfacil.com/ESO/logaritmos/slog9-03.png[/img][br][br]that has as solution:[br][br][img]https://www.matesfacil.com/ESO/logaritmos/slog9-04.png[/img][br][br]Finally, we undo the change of variable:[br][br][img]https://www.matesfacil.com/ESO/logaritmos/slog9-05.png[/img][br][br]We do not need to check the solutions.
[b]More examples:[br][br][/b][list][br][*][url=https://www.matesfacil.com/english/high/logarithms-resolved-equations-systems-demonstrations.html]Logarithmic equations and systems[/url][br][br][*][url=https://www.matesfacil.com/english/]Idex of math content[/url][br][/list]