Logarithmic and logarithmic functions and models

Go to Part A: Logistic functions and models

This tutorial: Part B: Logarithmic functions and models

(This topic is also in Section 2.4 in Finite Mathematics and Applied Calculus)

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Function evaluator and grapher

Excel grapher

Note To work with logarithmic functions, we need to understand what logarithms are, and also know the logarithm identities. If you are not comfortable with these requirements, visit the %%logstut. For a review on natural logarithms, visit the %%ExpFuncsB. Following is a warmup about logarithm identites, taken from the %%logstut:

Logarithmic functions: Basics

Logarithmic function

Logarithmic functions have the following form:

$f(x) = \log_b x + C$ \t \gap[10] $C$ %%and $b$ #[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
$\log_b(b^{-1}) = -\log_b(b) = -1$ \\ $\log_b(b^{-2}) = -2\log_b(b) = -2$ \\ $\log_b(b^{-3}) = -3\log_b(b) = -3$ \\ ... \\ $\log_b(b^{-n}) = -n\log_b(b) = -n.$

%%Examples

1. #[The Logarithmic function][La función logarítmica]# $f(x) = \log_2 x - 1$ #[has][tiene]# $b = 2$ %%and $C = -1.$ #[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:]#

2. #[One for you][Uno para ti]#

#[Application: Earthquakes][Aplicación: Temblores]#

#[The magnitude of a large earthquake can be measured by the formula][La magnitud de un gran terremoto se puede medir con la fórmula]#

$\displaystyle M = \frac{2}{3}\log S - 10.7,$ \gap[10] \t Moment magnitude seismic scale

where $S$ is the seismic moment, which measures the intensity of the earthquake based on the slippage and size of the fault and the rigidity of the fault material.^†

† $S$ is measured in ergs, which are units also used to measure energy. (An erg is the amount of energy it takes to accelerate a stationary mass of 2 grams to a velocity of 1 cm/sec.)

(a) Calculate the seismic moment of a 6.0 magnitude earthquake.
(b) The earthquake in Lice, Turkey on July 28 1976 had a magnitude of 6.0 and the earthquake in Kerman, Iran on December 26 2003 had a magnitude of 6.6. Compare the two: The seismic moment in the Turkey earthquake was what percentage of the seismic moment of the Iran quake?

#[Solution][Solutión]#

(a)

$M = \dfrac{2}{3}\log S - 10.7$ \\ $6.0 = \dfrac{2}{3}\log S - 10.7$

We need to solve this equation for $M_0$.:

$\dfrac{2}{3}\log S = 6.0 + 10.7 = 16.7$ \gap[20] \t Move the 10.7 over. \\ $\log S = \dfrac{3}{2}(16.7) = 25.05$ \gap[20] \t Multiply by $3/2$. \\ $S = 10^{25.05} \approx 1.12 \times 10^{25}$ #[ergs][ergios]#. \gap[20] \t Exponent form

(b) #[Take $S_1$ to be the seismic moment in the Iran quake and $S_2$ to be the moment in the Turkey quake. The calculation we just did can be written as][Toma $S_1$ como el momento sísmico del terremoto de Irán y $S_2$ como el momento del terremoto de Turquía. El cálculo que acabamos de hacer se puede escribir como]#

$M_1 = 10^{(3/2)(\color{indianred}{6.0} + 10.75)}$ \t #[Turkey][Turquía]# \\ $M_2 = 10^{(3/2)(\color{indianred}{6.6} + 10.75)}$ \t #[Iran][Irán]# \\ $\dfrac{S_1}{S_2} = 10^{(3/2)(\color{indianred}{6.0} + 10.75) -(3/2)(\color{indianred}{6.6} + 10.75)} \qquad$ \t #[Laws of exponents][Leyes de los exponentes]# \\ \gap[20] ${}= 10^{(3/2)(\color{indianred}{6.0 - 6.6})}$ \t #[Simplify][Simplifica]# \\ \gap[20] ${}= 10^{(3/2)(-0.6)} = 10^{-0.9} \approx 0.13$

#[Thus, the seismic moment in the Turkey quake was about 13% of that in the Iran quake.][Por lo tanto, el momento sísmico en el terremoto de Turquía fue aproximadamente el 13% del del terremoto de Irán.]#

Now try some of the exercises in Section 2.4 in Finite Mathematics and Applied Calculus. or move ahead by pressing "Next tutorial" on the sidebar.