Saltar al contenido principal

# 3.5: Cambio de Base

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$

( \newcommand{\kernel}{\mathrm{null}\,}\) $$\newcommand{\range}{\mathrm{range}\,}$$

$$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$

$$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$

$$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$

$$\newcommand{\Span}{\mathrm{span}}$$

$$\newcommand{\id}{\mathrm{id}}$$

$$\newcommand{\Span}{\mathrm{span}}$$

$$\newcommand{\kernel}{\mathrm{null}\,}$$

$$\newcommand{\range}{\mathrm{range}\,}$$

$$\newcommand{\RealPart}{\mathrm{Re}}$$

$$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$

$$\newcommand{\Argument}{\mathrm{Arg}}$$

$$\newcommand{\norm}[1]{\| #1 \|}$$

$$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$

$$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\AA}{\unicode[.8,0]{x212B}}$$

$$\newcommand{\vectorA}[1]{\vec{#1}} % arrow$$

$$\newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow$$

$$\newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vectorC}[1]{\textbf{#1}}$$

$$\newcommand{\vectorD}[1]{\overrightarrow{#1}}$$

$$\newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}}$$

$$\newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}}$$

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

$$\newcommand{\avec}{\mathbf a}$$ $$\newcommand{\bvec}{\mathbf b}$$ $$\newcommand{\cvec}{\mathbf c}$$ $$\newcommand{\dvec}{\mathbf d}$$ $$\newcommand{\dtil}{\widetilde{\mathbf d}}$$ $$\newcommand{\evec}{\mathbf e}$$ $$\newcommand{\fvec}{\mathbf f}$$ $$\newcommand{\nvec}{\mathbf n}$$ $$\newcommand{\pvec}{\mathbf p}$$ $$\newcommand{\qvec}{\mathbf q}$$ $$\newcommand{\svec}{\mathbf s}$$ $$\newcommand{\tvec}{\mathbf t}$$ $$\newcommand{\uvec}{\mathbf u}$$ $$\newcommand{\vvec}{\mathbf v}$$ $$\newcommand{\wvec}{\mathbf w}$$ $$\newcommand{\xvec}{\mathbf x}$$ $$\newcommand{\yvec}{\mathbf y}$$ $$\newcommand{\zvec}{\mathbf z}$$ $$\newcommand{\rvec}{\mathbf r}$$ $$\newcommand{\mvec}{\mathbf m}$$ $$\newcommand{\zerovec}{\mathbf 0}$$ $$\newcommand{\onevec}{\mathbf 1}$$ $$\newcommand{\real}{\mathbb R}$$ $$\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}$$ $$\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}$$ $$\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}$$ $$\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}$$ $$\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$$ $$\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$$ $$\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$$ $$\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$$ $$\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}$$ $$\newcommand{\laspan}[1]{\text{Span}\{#1\}}$$ $$\newcommand{\bcal}{\cal B}$$ $$\newcommand{\ccal}{\cal C}$$ $$\newcommand{\scal}{\cal S}$$ $$\newcommand{\wcal}{\cal W}$$ $$\newcommand{\ecal}{\cal E}$$ $$\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}$$ $$\newcommand{\gray}[1]{\color{gray}{#1}}$$ $$\newcommand{\lgray}[1]{\color{lightgray}{#1}}$$ $$\newcommand{\rank}{\operatorname{rank}}$$ $$\newcommand{\row}{\text{Row}}$$ $$\newcommand{\col}{\text{Col}}$$ $$\renewcommand{\row}{\text{Row}}$$ $$\newcommand{\nul}{\text{Nul}}$$ $$\newcommand{\var}{\text{Var}}$$ $$\newcommand{\corr}{\text{corr}}$$ $$\newcommand{\len}[1]{\left|#1\right|}$$ $$\newcommand{\bbar}{\overline{\bvec}}$$ $$\newcommand{\bhat}{\widehat{\bvec}}$$ $$\newcommand{\bperp}{\bvec^\perp}$$ $$\newcommand{\xhat}{\widehat{\xvec}}$$ $$\newcommand{\vhat}{\widehat{\vvec}}$$ $$\newcommand{\uhat}{\widehat{\uvec}}$$ $$\newcommand{\what}{\widehat{\wvec}}$$ $$\newcommand{\Sighat}{\widehat{\Sigma}}$$ $$\newcommand{\lt}{<}$$ $$\newcommand{\gt}{>}$$ $$\newcommand{\amp}{&}$$ $$\definecolor{fillinmathshade}{gray}{0.9}$$

Si bien es posible cambiar bases volviendo siempre a la forma exponencial, es más eficiente averiguar cómo cambiar la base de logaritmos en general. ya que solo hay logaritmos base$$e$$ y base 10 en la mayoría de las calculadoras, ¿cómo evaluarías una expresión como$$\log _{3} 12$$?

## Cambio de la base de logaritmos

El cambio de propiedad base establece:

$$\log _{b} x=\frac{\log _{a} x}{\log _{a} b}$$

Puede derivar esta fórmula convirtiendo$$\log _{b} x$$ a forma exponencial y luego tomando la base logarítmica$$x$$ de ambos lados. Esto se muestra a continuación.

\begin{aligned} \log _{b} x &=y \\ b^{y} &=x \\ \log _{a} b^{y} &=\log _{a} x \\ y \log _{a} b &=\log _{a} x \\ y &=\frac{\log _{a} x}{\log _{a} b} \end{aligned}

Por lo tanto,$$\log _{b} x=\frac{\log _{a} x}{\log _{a} b}$$

Si tuvieras que evaluar$$\log _{3} 4$$ usando tu calculadora, es posible que necesites usar el cambio de fórmula base ya que algunas calculadoras solo tienen base 10 o base$$e$$. El resultado sería:

$$\log _{3} 4=\frac{\log _{10} 4}{\log _{10} 3}=\frac{\ln 4}{\ln 3} \approx 1.262$$

## Ejemplos

### Ejemplo 1

Anteriormente, se le preguntó cómo usar una calculadora para evaluar una expresión como$$\log _{3} 12$$. Para evaluar una expresión como$$\log _{3} 12$$ tienes algunas opciones en tu calculadora:

$$\frac{\ln 12}{\ln 3}=\frac{\log 12}{\log 3} \approx 2.26$$

Algunas calculadoras gráficas también tienen otra opción. Presiona el MATH seguido de los botones A e ingresa$$\log _{3} 12$$

### Ejemplo 2

Demostrar la siguiente identidad de registro.

$$\log _{a} b=\frac{1}{\log _{b} a}$$

$$\log _{a} b=\frac{\log _{x} b}{\log _{x} a}=\frac{1}{\frac{\log _{x} a}{\log _{x} b}}=\frac{1}{\log _{b} a}$$

### Ejemplo 3

Simplifique a un resultado exacto:$$\left(\log _{4} 5\right) \cdot\left(\log _{3} 4\right) \cdot\left(\log _{5} 81\right) \cdot\left(\log _{5} 25\right)$$

$$\frac{\log 5}{\log 4} \cdot \frac{\log 4}{\log 3} \cdot \frac{\log 3^{4}}{\log 5} \cdot \frac{\log 5^{2}}{\log 5}=\frac{\log 5}{\log 4} \cdot \frac{\log 4}{\log 3} \cdot \frac{4 \cdot \log 3}{\log 5} \cdot \frac{2 \cdot \log 5}{\log 5}=4 \cdot 2=8$$

### Ejemplo 4

Evaluar:$$\log _{2} 48-\log _{4} 36$$

\begin{aligned} \log _{2} 48-\log _{4} 36 &=\frac{\log 48}{\log 2}-\frac{\log 36}{\log 4} \\ &=\frac{\log 48}{\log 2}-\frac{\log 6^{2}}{\log 2^{2}} \\ &=\frac{\log 48}{\log 2}-\frac{2 \cdot \log 6}{2 \cdot \log 2} \\ &=\frac{\log 48-\log 6}{\log 2} \\ &=\frac{\log \left(\frac{48}{6}\right)}{\log 2} \\ &=\frac{\log 8}{\log 2} \\ &=\frac{\log 2^{3}}{\log 2} \\ &=\frac{3 \cdot \log 2}{\log 2} \\ &=3 \end{aligned}

### Ejemplo 5

Dado$$\log _{3} 5 \approx 1.465$$ hallazgo$$\log _{25} 27$$ sin usar un botón de registro en la calculadora.

$$\log _{25} 27=\frac{\log 3^{3}}{\log 5^{2}}=\frac{3}{2} \cdot \frac{1}{\left(\frac{\log 5}{\log 3}\right)}=\frac{3}{2} \cdot \frac{1}{\log _{3} 5} \approx \frac{3}{2} \cdot \frac{1}{1.465}=1.0239$$

Revisar

1. $$\log _{6} 15$$

2. $$\log _{9} 12$$

3. $$\log _{5} 25$$

4. $$\log _{8}\left(\log _{4}\left(\log _{3} 81\right)\right)$$

5. $$\log _{2} 3 \cdot \log _{3} 4 \cdot \log _{6} 16 \cdot \log _{4} 6$$

6. $$\log 125 \cdot \log _{9} 4 \cdot \log _{4} 81 \cdot \log _{5} 10$$

7. $$\log _{5}\left(5^{\log _{5} 125}\right)$$

8. $$\log \left(\log _{6}\left(\log _{2} 64\right)\right)$$

9. $$10^{\log _{100} 9}$$

10. $$\left(\log _{4} x\right)\left(\log _{x} 16\right)$$

11. $$\log _{49} 49^{5}$$

12. $$3 \log _{24} 24^{8}$$

13. $$4^{\log _{2} 3}$$

Demostrar las siguientes propiedades de logaritmos.

14. $$\left(\log _{a} b\right)\left(\log _{b} c\right)=\log _{a} c$$

15. $$\left(\log _{a} b\right)\left(\log _{b} c\right)\left(\log _{c} d\right)=\log _{a} d$$

This page titled 3.5: Cambio de Base is shared under a CK-12 license and was authored, remixed, and/or curated by CK-12 Foundation via source content that was edited to the style and standards of the LibreTexts platform.