Since a logarithm is defined as the inverse of an exponential equation, we can use this relationship to change between exponential and logarithmic forms:

\displaystyle y=\text{log}_b \left(x\right) \iff b^y=x

\bm{b}

is the base and b>0, b\neq 1

This relationship means that a logarithm of the form \text{log}_b \left(x\right) is equal to the exponent, y, to which we would raise the base, b, in order to obtain the argument, x.

When the logarithm is in the form y=\text{log}_bx, it is sometimes referred to as a common logarithm. When the base of a common logarithm is missing, the understood base is 10. \log x=\text{log}_{10}x

Logarithmic function

A function that contains a variable expression inside a logarithm; a function of the form f\left(x\right)=\text{log}_b\left(x\right), where x>0 and b>0, b\neq 1.

Natural logarithms are logarithms with a base of the mathematical constant e. The natural logarithmic function is y=\text{log}_e x which is commonly written as y=\ln x. The base of natural log is always e, so it is never written as a subscript.

To transform between exponential and logarithmic forms of a natural logarithm, we can use the definition of a natural logarithm:y=\ln \left(x\right) \iff e^y=x

Examples

Example 1

Rewrite the following logarithmic equations in exponential form.

y=\ln \left(x-3\right)

Worked Solution

Create a strategy

The understood base of a natural logarithm is e. We can use the property \ln \left(x\right)=y \iff x=e^y to rewrite it in exponential form.

Apply the idea

The equation in exponential form is e^y=x-3

Reflect and check

The base of the logarithm is always the same as the base of the exponent. For natural logarithms, the base is always e.

y=\log\left(\dfrac{1}{10}\right)

Worked Solution

Create a strategy

Whenever the base of a common logarithm is not written, it is understood to be 10. We can use the property \text{log}_b \left(x\right)=n \iff x=b^n with b=10, x=\dfrac{1}{10}, and n=y.

Apply the idea

This equation in exponential form is 10^{y}=\dfrac{1}{10}

Reflect and check

Now that is it in exponential form, it may be easier to recognize that y=-1 makes this equation true. 10^{-1}=\dfrac{1}{10} by the negative exponent property.

Example 2

Evaluate \text{log}_\frac{1}{3} 9 .

Worked Solution

Create a strategy

To evaluate the logarithm, we need to set it equal to a variable and rewrite the equation in exponential form. Then, we can solve for the variable.

Apply the idea

Let \text{log}_{\frac{1}{3}}9=y. We can rewrite this equation as 9=\left(\dfrac{1}{3}\right)^y

To solve for y, we want to find the exponent that we would raise \dfrac{1}{3} by, to obtain a result of 9. To do this, we want to rewrite 9 and \dfrac{1}{3} as exponential expressions with the same base. We can see that 9=3^2 and \left(\dfrac{1}{3}\right)^y=\dfrac{1}{3^y}=3^{-y}.

\displaystyle 9	\displaystyle =	\displaystyle \left(\dfrac{1}{3}\right)^y	Original equation
\displaystyle 3^2	\displaystyle =	\displaystyle 3^{-y}	Rewrite both sides as a power of 3
\displaystyle 2	\displaystyle =	\displaystyle -y	If a^m=a^n, then m=n.
\displaystyle -2	\displaystyle =	\displaystyle y	Divide by -1

Therefore, \text{log}_{\frac{1}{3}}9=-2

Reflect and check

Any time the base is a fraction and the argument is a whole number or vice versa, the negative exponent property will be used. This means the result will always be a negative value.

Example 3

Evaluate \log_{8} \left(\dfrac{1}{64}\right).

Worked Solution

Create a strategy

Use the definition: 'If y=\log_{b}x then b^y=x.'

Apply the idea

\displaystyle \log_{8} \left(\dfrac{1}{64}\right)	\displaystyle =	\displaystyle y	Write the equation
\displaystyle 8^y	\displaystyle =	\displaystyle \dfrac{1}{64}	Write in exponential form
\displaystyle 8^y	\displaystyle =	\displaystyle \dfrac{1}{8^2}	Write the denominator as a power of 8
\displaystyle 8^y	\displaystyle =	\displaystyle 8^{-2}	Use the negative exponent property
\displaystyle y	\displaystyle =	\displaystyle -2	Equate the exponents
\displaystyle \log_{8} \left(\dfrac{1}{64}\right)	\displaystyle =	\displaystyle -2

Example 4

Evaluate \log_{10}45. Round your answer to two decimal places.

Worked Solution

Create a strategy

Use a calculator or other technology to find a decimal value for the logarithm.

Apply the idea

Press the log button and enter the number 45 to get:\log_{10}45 \approx 1.65

Idea summary

A logarithm is defined as the inverse of an exponential equation. We can convert between exponential and logarithmic forms using the property y=\text{log}_b \left(x\right) \iff b^y=x for common logarithms or y=\ln \left(x\right) \iff e^y=x for natural logarithms.

If the base of a common logarithm is not written, the understood base is 10. The base of a natural logarithm is always e.

Logarithmic scales

Logarithmic scales can be useful when comparing numbers that have a very large range. If comparing the distance of the planets from the sun you could use a logarithmic scale to simplify the numbers.

\text{Planet}	\text{Distance from the sun, } d \text{ (km)}	\log_{10} \left(d\right)
\text{Mercury}	57 \,910 \,000	7.76
\text{Venus}	108 \,200 \,000	8.03
\text{Earth}	149 \,600 \,000	8.17
\text{Mars}	227 \,900 \,000	8.36
\text{Jupiter}	778 \,500 \,000	8.91

Consider the difference between Mercury and Jupiter's distances from the sun. By observation you see that the difference in the \text{log} values as a little bit more than 1 so Jupiter's distance from the sun is a little more than 10 times the distance of Mercury. To find a more exact answer:

\displaystyle 8.91-7.76	\displaystyle =	\displaystyle 1.15	Find the difference between Jupiter and Mercury
\displaystyle 10^{1.15}	\displaystyle =	\displaystyle 14.13	Write 1.15 as a power of 10 and evaluate

So you could say that Jupiter is 14 times the distance of Mercury from the sun.

Logarithmic scales become more useful when comparing very large and very small numbers, like comparing the distance a planet is from the sun with the distance between say your house and your school. That might be interesting but not a very practical comparison.

On a log scale every 1 unit up means you are increasing by a factor of 10, every 2 increases by a factor of 10^2 = 100.

Logarithm scales are often used when there is a large range of values involved with the variables under consideration. Here is an example to motivate the idea of a log scale.

Consider the set of five ordered pairs shown here:

x	1	2	3	4	5
y	20	200	3631	52\,481	250\,000

A plot of the five points would be difficult to manage because of the range of the y values.

See the following graph plot, the scale on the y-axis is so huge, that we lost a lot of the information from the first 3 points.

50000

100000

150000

200000

250000

Some of the data is lost due to the scale so we could plot the base 10 logarithm of y against x, with values shown in a new table.

x	1	2	3	4	5
\log _{10}y	1.301	2.301	3.560	4.720	5.398

Even though the y values are far more manageable in this form, we need to remember that the actual data points are those in the first table. That is to say the actual y values, correct to 3 decimal places at least, are given by 10^{1.301},10^{2.301},10^{3.560},10^{4.720}, and 10^{5.398}.

Using the logarithm of the y values gives us the following graph.

Measuring the size of earthquakes

One common practical use is measuring the intensity of earthquakes. Since the 1930's the Richter magnitude scale has been used to describe the intensity or 'size' of an earthquake.

Due to the number of variables involved in measuring an earthquake its value on the Richter scale is calculated using the log scale, the size of the seismic wave recorded and the distance the recording was from the source of the earthquake.

The value on the Richter scale indicates the intensity and can be compared with others using powers of 10 to describe how much stronger it was to another. The table below shows the intensity strength of an earthquake of particular intensity when compared to another earthquake.

\text{Earthquake Size}	3	4	5	6	7
3	0	+10	+100	+1000	+10\,000
4	-10	0	+10	+100	+1000
5	-100	-10	0	+10	+100
6	-1000	-100	-10	0	+10
7	-10\,000	-1000	-100	-10	0

Measuring level of sound

Another commonly known practical example of this is when measuring sound levels. Sound levels are a measure of the intensity of the sound waves on your eardrum. In this case, the sound intensity measure is compared to a reference intensity of 10^{-12} using a ratio.

The numbers given by the decibel scale indicate an intensity 10 times stronger for every 10 \text{ dB}. The table below shows the difference in intensity strength of a sound of a particular size when compared to another sound.

\text{Decibels}	50	60	70	80	90
50	0	+10	+100	+1000	+10\,000
60	-10	0	+10	+100	+1000
70	-100	-10	0	+10	+100
80	-1000	-100	-10	0	+10
90	-10\,000	-1000	-100	-10	0

Examples

Example 5

Below is a table of values that shows a \text{log} scale relating x and y. Form an equation relating x and y. Express the equation in logarithmic form.

\text{log scale measure ($y$)}		\text{linear measure ($x$)}
0	=	1
1	=	10
2	=	100
3	=	1000
4	=	10\,000

Worked Solution

Create a strategy

Consider the pattern in each row and express the equation into b=\log _am.

Apply the idea

Notice that each x value increases by 10 so the base power is 10.

Also, when 10^0 where y=0 is equal to x=1.

10^0=1

Substituting 0=y and 1=x gives us the equation 10^y=x.

Recall that a^b=m can be rearranged into b=\log _am.

\displaystyle b	\displaystyle =	\displaystyle \log _am	Write the logarithmic formula
\displaystyle y	\displaystyle =	\displaystyle \log _{10}x	Subsitute a=10, b=y,m=x

Example 6

The histogram below shows the area (in \text{ km}^2) for 12 countries, plotted using a log scale.

This image shows a histogram of land areas and number of countries. Ask your teacher for more information.

How many countries have an area of between 10\,000 \text{ km}^2 and 100\,000 \text{ km}^2?

Worked Solution

Create a strategy

Count the number of 0s of each number and find their corresponding value in the vertical axis.

Apply the idea

10\,000 has 4 zeros while 100\,000 has 5 zeros.

We can see from the histogram that 4-5 is corresponding to 3 in the vertical axis.

So there are 3 countries with an area between 10\,000 \text{ km}^2 and 100\,000 \text{ km}^2.

Example 7

The Richter Scale is a base-10 logarithmic scale used to measure the magnitude of an earthquake, given by R=\log _{10}x, where x is the relative strength of the quake. This means an earthquake that measures 4.0 on the Richter Scale will be 10 times stronger than one that measures 3.0.

The aftershock of an earthquake measured 6.7 on the Richter Scale, and the main quake was 4 times stronger. Solve for r, the magnitude of the main quake on the Richter Scale, to one decimal place.

Worked Solution

Create a strategy

Follow the steps to solve for r.

Write R=\log _{10}x as a power of 10. Multiply x by the strength.
Set an equation for the aftershock strength.
Equate the values of x.

Apply the idea

\displaystyle R	\displaystyle =	\displaystyle \log _{10}x	Write the Richter Scale formula
\displaystyle x	\displaystyle =	\displaystyle 10^R	Write as a power of 10
\displaystyle x	\displaystyle =	\displaystyle 4\times10^R	Multiply by the strength

Since x is the relative strength, we can say that the equation for aftershock strength is x=10^r.

Both equations has the same left-hand signs, x. Then we can equate their right-hand sides.

\displaystyle 10^r	\displaystyle =	\displaystyle 4\times10^R	Equate the values of x
\displaystyle 10^r	\displaystyle =	\displaystyle 4\times10^{6.7}	Substitute R=6.7
\displaystyle \dfrac{10^r}{10^{6.7}}	\displaystyle =	\displaystyle \dfrac{4\times10^{6.7}}{10^{6.7}}	Divide both sides by 10^{6.7}
\displaystyle \dfrac{10^r}{10^{6.7}}	\displaystyle =	\displaystyle 4	Evaluate
\displaystyle 10^{r-6.7}	\displaystyle =	\displaystyle 4	Apply \dfrac{a^m}{a^n}=a^{m-n}
\displaystyle \log\left(10^{r-6.7}\right)	\displaystyle =	\displaystyle \log 4	Take the logarithm of both sides
\displaystyle r-6.7	\displaystyle =	\displaystyle \log 4	Apply \log _aa^n=n
\displaystyle r-6.7+6.7	\displaystyle =	\displaystyle \log 4+6.7	Add 6.7 to both sides
\displaystyle r	\displaystyle =	\displaystyle 7.3	Evaluate

Example 8

The decibel scale, which is used to record the loudness of sound, is a logarithmic scale.

In the decibel scale, the lowest audible sound, with intensity 10^{-12} \text{ watts/m$^2$} is assigned the value of 0.
A sound that is 10 times louder is assigned a decibel value of 10.
A sound 100 (10^2) times louder is assigned a decibel value of 20.
A sound 1000 (10^3) times louder is assigned a decibel value of 30.

If the sound of a normal speaking voice is 50 decibels, and the sound in a bus terminal is 80 decibels, then how many times louder is the bus terminal compared to the speaking voice?

Give your final answer as a basic numeral, not in exponential form.

Worked Solution

Create a strategy

Subtract the \text{dB} of normal speaking voice from the \text{dB} of bus terminal sound.

Apply the idea

\displaystyle \text{Decibel value}	\displaystyle =	\displaystyle 80-50	Subtract 50\text{ dB} from 80\text{ dB}
	\displaystyle =	\displaystyle 30 \text{ dB}	Evaluate

The given states that a sound 1000 (10^3) times louder is assigned a decibel value of 30.

So the bus terminal sound is 1000 louder than the speaking voice.

Idea summary

Logarithmic scales are useful to compare very large and very small numbers.

Instead of the actual very large or very small numbers, we can use their logarithmic scales to use as values on a graph for better representation.

We can use logarithms to the real-life situations where very large or very small numbers are involved such as measuring the earthquake sizes and level of sound.

2.9 Logarithmic expressions

Introduction

Ideas

Logarithmic expressions

Examples

Example 1

Example 2

Example 3

Example 4

Logarithmic scales

Examples

Example 5

Example 6

Example 7

Example 8

Outcomes

2.9.A

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