Common Logarithms

• If x is a positive number, log x is the exponent of 10 that gives x. That is, y = log x if and only if 10y = x.

• The function log x is known as the common logarithm function or, simply, the log function. It may also be written as log10 x, which is read as “log to the base 10 of x”.

• Example. What is log 100? Since 102 = 100, we have log 100 = 2.

What is log 0.01? Since 10-2 = 0.01, we have log 0.01 = -2.

What is Since we have ?10 log , 1010 5.0 . 0.5 10 log

• Two important values of the log function are:

• By definition, the log is the inverse function of the exponential with base 10, so we have:

• For a and b both positive and any value of t,

1. 10 log and 0 1 log

x,allfor x,10 log x

0.for x x,10 xlog

b log a log ab)( log

b log a log b

a log

. b log t )blog( t

How many years will it take for your salary to double?

• Problem. If you start at $40000, and you are given a 6% raise each year, how many years must pass before your salary is at least $80000?

Solution. We must solve (1.06)t(40000) = 80000 for t.

Equivalently, we must solve (1.06)t = 2 for t. If we take the log of both sides of this equation and use the last property on the previous slide, we obtain

If you have to wait until the end of the year to actually get your raise, 12 years must pass.

or 2, log 1.06 logt

years. 896.11025306.0

30103.0

1.06 log

2 log t

Half-life of the dotcom investment

• How many years must pass before the investment discussed previously has a value which is one-half of its original value? We must solve the following equation for t:

becomes which 5000, )10000()95.0( t

10000.by dividingupon 0.5, )95.0( t

logs. ngafter taki obtained is 0.5 log 95.0 logt

years. 51.13022276.0

30103.0

0.95 log

0.5 log t

Graphical Solution to dotcom half-life

(13.51, 5000)

Solving a logarithmic equation

• Solve for x:

log(2x+1) = 3

10log(2x+1) = 103 exponentiate

2x+1 = 1000 evaluate

2x = 999 subtract

x = 499.5 divide

Fallacies Involving Logs

• log(a + b) is not the same as log a + log b .

• log(a – b) is not the same as log a – log b .

• log(ab) is not the same as (log a)(log b) .

• .

• .

b log

a log as same thenot is

b

alog

a log

1 as same thenot is

a

1log

Natural Logarithms

• If x is a positive number, ln x is the exponent of e that gives x. That is, y = ln x if and only if ey = x.

• The function ln x is known as the natural logarithm function. It may also be written as loge x, which is read as “log to the base e of x”.

• Example. What is ln e? Since e1 = e, it follows that ln e = 1.

What is ln (1/e)? Since e-1 = 1/e, it follows that ln (1/e) = -1.

What is Since it follows that ?eln , ee 5.0 . 0.5 eln

• Two important values of the ln function are:

• By definition, ln is the inverse function of the exponential with base e, so we have:

• For a and b both positive and any value of t,

1. eln and 0 1ln

x,allfor x,eln x

0.for x x,e ln x

bln aln ab)(ln

bln aln b

aln

. bln t bln t

Converting Between Q = abt and Q = aekt

• Any exponential function can be written in either of two forms: Q = abt and aekt.

• Problem. Convert the exponential function P = 175(1.145)t to the form P = aekt. Solution. We must find k such that

Therefore,

b.ln k then ,e b If k

0.1354. 1.145ln k

or ,145.1ek

.175e P 0.1354t

Logarithms and exponential models

• Problem 47, Section 4.2. Suppose the temperature H, in °F, of a cup of coffee t hours after it is set out to cool is given by the equation:

How long does it take the coffee to cool down to 90°F?

Solution. We must solve the following equation for t:

.120(1/4) 70 H t

90, 120(1/4) 70 t gsubtractinby 20, 120(1/4) t

logs by taking log(1/6), log(1/4)t dividingby 1/6, (1/4) t

property log a using log(1/6), log(1/4)t hours. 1.29og(1/4)log(1/6)/l t

Some problems require graphical methods for solution

• Problem. Find the positive value of x which satisfies:

If we take the log of both sides, we get an equation with a logarithm term and we are unable to isolate x. Solution. Use graphical approach:

2. x 10x

The relation between ln x and log x

• We start with the equation

and we apply ln to both sides of this equation. We have

Next, we apply a property from a previous slide to get

The last equation tells us that ln x equals log x times the constant factor, ln 10 = 2.3026. Because of this fact, the shapes of the graphs of ln x and log x are similar.

ln x. 10ln xlog

x10 xlog

ln x. 10ln x log

The graph, domain, and range of the common logarithm

• It follows from the definition of log x that its domain consists of all positive real numbers. Its range is all real numbers.

• Using Maple or graphing calculator, we can plot the graph of log x:

x 0.01 0.1 1 10 100 1000

log x -2 -1 0 1 2 3

The graphs of 10x and log x using Maple

> plot({10^x,x,piecewise(x>0,log10(x))},x=-4..10,-4..10,color=black,

scaling=constrained);

More Detail for Graphs of 10x and log(x)

(1,0)

(0,1)

(0.3010,2)

(2,0.3010)

·

·

(0.1,–1)·

·(–1,0.1)

Asymptote for log x

• The exponential function 10x has the property that its graph approaches the x-axis as x gets large and negative. We recall that this behavior is described by saying that 10x has the x-axis as a horizontal asymptote. We can also express this by writing or by

• Corresponding to the above property for the exponential is the property that the graph of log x approaches the negative y-axis as x approaches 0 from the right. This behavior is described by saying that log x has the negative y-axis as a vertical asymptote. We can also express this by writing

or by

.010lim x

x

xas 010x

0 xas xlog . xloglim0x

Chemical Acidity

• In chemistry, the acidity of a liquid is expressed using pH. The acidity depends on the hydrogen ion concentration in the liquid (in moles per liter). This concentration is written [H+]. The pH is defined as:

• Problem. A vinegar solution has a pH of 3. Determine the hydrogen ion concentration.

Solution. Since 3 = – log[H+], we have –3 = log[H+]. This means that 10-3 = [H+]. The hydrogen ion concentration is 10-3 moles per liter.

].[H log pH

Logarithms and orders of magnitude

• We often compare sizes or quantities by computing their ratios. If A is twice as tall as B, then

Height of A/Height of B = 2.

• If one object is 10 times heavier than another, we say it is an order of magnitude heavier. If one quantity is two factors of 10 greater than another, we say it is two orders of magnitude greater, and so on.

• Example. The value of a dollar is two orders of magnitude greater than the value of a penny.

We note that the order of magnitude is the logarithm of the ratio of their values.

.1001.0$

1$ 2

• To measure a sound in decibels, the sound’s intensity, I, in watts/cm2 is compared to a standard benchmark sound, I0. This results in the following definition:

where I0 is defined to be 10-16 watts/cm2, roughly the lowest intensity audible to humans.

• Problem. If a sound doubles in intensity, by how many units does its decibel rating increase?

Decibels

,I

Ilog 10

0

decibelsinlevelNoise

00 I

Ilog10

I

2Ilog 10 ratings decibelin Difference

dB. 010.32log 10 I

Ilog

I

2Ilog10

00

Summary for Logarithmic Functions

• log x is the exponent of 10 that gives x. The log function is the inverse function of the exponential function with base 10.

• ln x is the exponent of e that gives x. The ln function is the inverse function of the exponential function with base e.

• log x and ln x have several “properties of logarithms”.

• Using logarithms, we can solve certain equations involving exponential functions.

• The domain of the logarithm is all positive numbers, and the range of the logarithm is all real numbers.

• The graphs of the logarithm and the exponential function (with the same base) are “mirror images” across the line y = x.

• The graph of the logarithm has the y-axis as a vertical asymptote.

• Logarithms are used to measure quantities which vary over a wide range. E. g., hydrogen ion concentration & sound intensity.