Logarithmic function

Logarithmic functions have the following form:

$f(x) = \log_b x + C$ \t \gap[10] $C$ #[and][y]# $b$ #[are arbitrary constants with $b$ positive and not equal to 1.][son constantes arbitrarias con $b$ positivo y no igual a 1.]# \\ #[Technology formula][Fórmula tecnológica]#: \t \gap[10] log(x)/log(b)+C %%or ln(x)/ln(b)+C

#[Alternative forms][Formas alternativas]#

We can use the logarithm identities from the %%logstut write $\log_b x$ in the form

\t $\dfrac{\log x}{\log b} = \dfrac{1}{\log b}\log x = A\log x$ \t $A = \dfrac{1}{\log b} ={}$ #[constant][constante]# \\ #[or][o]# \t $\dfrac{\ln x}{\ln b} = \dfrac{1}{\ln b}\ln x = A\ln x$ \t $A = \dfrac{1}{\ln b} ={}$ #[constant][constante]#

giving us two alternative forms of a general logarithmic function

$f(x) = A\log x + C$ \\ $f(x) = A\ln x + C$ \t $\ln x = \log_e x$ is the natural logarithm. See %%ExpFuncsB for more on the number $e$ and the natural logarithm.

where $A$ and $C$ are arbitrary constants with $A \ne 0$.

Their graphs have the form shown below. When $b \gt 1$ $f(x)$ increases with increasing $x$, and when $b \lt 1$ it decreases with increasing $x$:

$\bold{f(x) = \log_b x}$ \t \\

$f(x) = \log_b x$ #[Increasing when ][Aumentando cuando ]# $b \gt 1$

$f(x) = \log_b x$ #[Decreasing when ][Disminuyendo cuando ]# $b \lt 1$

Adding a constant $C$ has the effect of shifting the graphs vertically $C$ units:

$\bold{f(x) = \log_b x + C}$ \t \\

$f(x) = \log_b x + C$ #[Increasing when ][Aumentando cuando ]# $b \gt 1$

$f(x) = \log_b x + C$ #[Decreasing when ][Disminuyendo cuando ]# $b \lt 1$

#[Some features seen in the above graphs][Algunas características que se ve en las gráficas anteriores]#

As $\log_b 1 = 0$, the graph of $\log_b x$ crosses the $x$-axis at $x = 1.$
If $b \gt 1$, the values $x = b, b^2, b^3, ...$ form an increasing sequence and the corresponding values of $\log_b x$ are
$\log_b(b) = 1$ \\ $\log_b(b^2) = 2\log_b(b) = 2$ \\ $\log_b(b^3) = 3\log_b(b) = 3$ \\ ... \\ $\log_b(b^n) = n\log_b(b) = n.$
If $b \lt 1$, the values $x = b^{-1}, b^{-2}, b^{-3}, ...$ form an increasing sequence and the corresponding values of $\log_b x$ are
$\log_b(b^{-1}) = -\log_b(b) = -1$ \\ $\log_b(b^{-2}) = -2\log_b(b) = -2$ \\ $\log_b(b^{-3}) = -3\log_b(b) = -3$ \\ ... \\ $\log_b(b^{-n}) = -n\log_b(b) = -n.$

#[Example][Ejemplo]#

#[The Logarithmic function][La función logarítmica]# $f(x) = \log_2 x - 1$ #[has][tiene]# $b = 2$ #[and][y]# $C = -1.$ #[as $b \gt 1$ the values of $f$ increase with increasing $x.$][ya que $b \gt 1$ los valores de $f$ aumentan al aumentar $x.$]# #[Here is a table calculating various values of $f$ and the resulting graph:][Aquí hay una tabla que calcula varios valores de $f$ y la gráfica resultante:]#

#[One for you][Alguno para ti]#

a. #[The %0 magnitude earthquake on %1 in %2 had a seismic moment of][El terremoto de magnitud %0 de %1 en %2 tuvo un momento sísmico de]#

$S = {}$ BOX $\times 10^{25}$ #[ergs][ergios]#

#[Accurate to two dignificant digits][Exacta hasta dos dígitos significados]#

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b. #[By comparison, the seismic moment of %3 magnitude earthquake on %4 in %5 was][En comparación, el momento sísmico del terremoto de magnitud %3 en %4 en %5 fue]#

BOX %7

#[Accurate to two dignificant digits][Exacta hasta dos dígitos significados]#

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