1044.sagews

Multiplication is repeated addition, and division is repeated subtraction, right?

So if finding powers is repeated multiplication, would finding logs be repeated ... division?

pow(2, 7)

1*2*2*2*2*2*2*2

log(128, 2)

128/2/2/2/2/2/2/2

Find the inverse of $f(x) = b^x$.

$y = b^x$

-------------------------

$x = b^y$

$y = \log_{b}x$

$g(x) = \log_{b}x$

So $f$ is the result of beginning with $1$ and multiplying by $b$ $x$ number of times,

and $g$ is the number of times, beginning with $x$, you must divide by $b$ to get back to $1$.

2^3

pow(2,3)

log(8, 2)

EXAMPLE 1  Changing from Logarithmic to Exponential Form

Write each equation in exponential form:

1a)  $2 = \log_{5}x

x == 5^2

pow(5, 2)

log(25, 5)

$y = b^x$

$y$ is the power, $b$ is the base, and $x$ is the exponent.  

A logarithm is an exponent!

Therefore, $x$ is also a log.  It is the $log$ of $y$.  $x$ is the exponent that relates $b$ to $y$.

If $y$ is the power of $b$ to the $x$, then $x$ is the log of $y$ base $b$.

$x = \log_{b}y$

var('y b x')

y == pow(b,x)

x == log(b,y)

1b)  $3 = \log_{b}64$

pow(b, 3) == 64

b^3 == 64

pow(64, 1/3)

1c)  $\log_{3}7 = y$

3^y == 7

(log(7)/log(3)).n(digits = 4)

EXAMPLE 2  Changing from Exponential to Logarithmic Form

Write each equation in log form:

2a)  $12^2 = x$

2 == log(x, 12)

solve(_, x)

2b)  $b^3 = 8$

3 == log(8, b)

2c)  $e^y = 9$

y == log(9, e)

EXAMPLE 3  Evaluating Logarithms

3a)  $\log_{2}16$

log(16, 2)

3b)  $\log_{7}\frac{1}{49}$

log(1/49, 7)

http://www.wolframalpha.com/input/?i=log(7%2C+1%2F49)

3c)  $\log_{36}6$

log(6, 36)

http://www.wolframalpha.com/input/?i=log(36%2C+6)

3d)  $log_3 \sqrt[7]{3}$

log(3^(1/7), 3)

http://www.wolframalpha.com/input/?i=log(3%2C+pow(3%2C+1%2F7))

EXAMPLE 4  Using Properties of Logs

4a)  $\log_7 7$

log(7, 7)

4b)  $\log_5 1$

log(1, 5)

EXAMPLE 5  Inverse Log Properties

5a)  $\log_4 4^5$

log(4^5, 4)

5b)  $6^{\log_6 9}

6^log(9, 6)

http://www.wolframalpha.com/input/?i=6^log(6%2C+9)

Log Properties

1.  $\log_b 1 = 0$

var('b x')

log(1, b)

2.  $\log_b b = 1$

log(b, b)

3.  $\log_b b^x = x$

log(b^x, b)

http://www.wolframalpha.com/input/?i=log(b%2C+b^x)

4.  $b^{\log_b x} = x$

b^log(x, b)

http://www.wolframalpha.com/input/?i=b^log(b%2C+x)

EXAMPLE 6  Graphs of Exponential and Log Functions

Graph $f(x) = 2^x$ and $g(x) = \log_2 x$.

f(x) = 2^x
g(x) = log(x, 2)

F = [(x, f(x)) for x in [-3..3]]
G = [(y, x) for (x, y) in F]

F

G

show(plot(f, -3, 3, color = 'purple') + plot(g, 1/8, 8, color = 'green') + \
points(F+G, rgbcolor = 'red') + \
line([(-3, -3), (8, 8)], rgbcolor = 'black'), aspect_ratio = 1)

EXAMPLE 7  Finding the Domain of a Log Function

Find the domain of $f(x) = \log_4 (x + 3)$.

f(x) = log(x+3, 4)

f(-3)

g(x) = log(x, 4)
f(x) = g(x+3)

show(plot(g, 0, 8, color = 'purple')+plot(f, -3, 5, color = 'green'), aspect_ratio = 1)

EXAMPLE 8  Modeling Heights of Children

$f(x) = 29 + 48.8\log(x+1)$ models adult height $f(x)$ of a boy who is $x$ years old. 

What % of his adult height has an 8 year old boy attained?

f(x) = 29 + 48.8*log(x+1, 10)

f(8).n()

EXAMPLE 9  Earthquake Intensity

$R = \log \frac{I}{I_0}$ gives magnitude $R$ of an earthquake of intensity $I$ where $I_0$ is the intensity of a zero level quake.

What was the magnitude of the 1906 San Francisco quake if it was $10^{8.3}$ times as intense as $I_0$?

log(10^8.3, 10)

EXAMPLE 10  Finding Domains of Natural Log Functions

Find each domain:

10a)  $f(x) = \ln (3 - x)$

f(x) = ln(3 - x)

f(3)

f(4)

f(2)

plot(f, -8, 3)

10b)  $h(x) = \ln(x - 3)^2$

h(x) = ln((x - 3)^2)

plot(h, -1, 7)

EXAMPLE 11  Heat

$f(x) = 13.4 \ln x - 11.6$ models temperature increase $f(x)$ after $x$ minutes.  Find the increase after 50 minutes.

f(x) = 13.4*ln(x) - 11.6

f(50).n()