3.4: Consejos y trucos de álgebra Parte VI (Logaritmos)
- Page ID
- 116626
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Un logaritmo es la función inversa a una función exponencial. Por ejemplo, para la función exponencial\(y = 2^x\), si tenemos una entrada de\(x = 6\), obtenemos una salida de\(y = 64\), y escribimos\(64 = 2^6\). La función logarítmica\(y = \log_2(x)\) es lo contrario de esto. Intercambiamos la entrada y la salida, así que ahora\(x = 64\) y\(y = 6\). Vemos\(6 = \log_2(64)\).
En el cálculo, utilizaremos mayormente la función exponencial\(e^x\) y su inversa,\(\ln(x)\). A continuación se presentan algunas fórmulas importantes:
\[\begin{align*} e^{\ln(x)} & = x \\ \ln(e^x) & = x \\ \ln(x) + \ln(y) & = \ln(xy) \\ \ln(x) - \ln(y) & = \ln\left(\frac{x}{y}\right) \\ a \ln(x) & = \ln(x^a) \end{align*}\]
Ejemplos:
Hay dos formas de hacer esta. Primero, podemos derribar al exponente de dos al frente\(\ln(x^2) = 2 \ln(x)\). Luego se pueden combinar los términos similares de\(2\ln(x)\) y\(\ln(x)\):
\[\begin{align*} \ln(x^2) - \ln(x) & = 2 \ln(x) - \ln(x) \\ & = \boxed{\ln(x)} \end{align*}\]
Alternativamente, podemos reescribir la resta como una división, así:
\[\begin{align*} \ln(x^2) - \ln(x) & = \ln\left(\frac{x^2}{x}\right) \\ & = \boxed{\ln(x)} \end{align*}\]
¡De cualquier manera obtenemos la misma respuesta!
Primero, reescribimos la multiplicación usando la suma. Entonces podemos simplemente a partir de ahí.
\[\begin{align*} \ln(e^3 x^4) - 3 \ln(x) & = \ln(e^3) + \ln(x^4) - 3 \ln(x)\\ & = 3 + 4 \ln(x) - 3 \ln(x) \\ & = \boxed{3 + \ln(x)} \end{align*}\]
Sabemos
, entonces.
Podemos reescribir todos los productos y divisiones como suma y resta:
\[\begin{align*} \ln\left(\frac{\sqrt{x} y}{z^3}\right) - \ln\left(\frac{z}{\sqrt{x} y^3}\right) & = \ln(\sqrt{x}) + \ln(y) - \ln(z^3) - [\ln(z) - \ln(\sqrt{x}) - \ln(y^3)] \\ & = \frac{1}{2} \ln(x) + \ln(y) - 3 \ln(z) - \ln(z) + \frac{1}{2} \ln(x) + 3 \ln(y) \\ & = \boxed{\ln(x) + 4 \ln(y) - 4 \ln(z)}. \end{align*}\]