© 2007  Rasmus ehf  and Jóhann Ísak Pétursson

Exponentials and logarithms

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Few numbers are as easy to work with as 1, 10, 100, 1000 e.t.c. All we have to do if we wish to multiply them together is count the number of zeros.

E.g. 10×100×1000 = 1 000 000 , a total of  6 zeros.

We can also see this if we write the numbers as powers of 10.

10×100×1000 = 10¹×10²×10³ = 10¹⁺²⁺³ = 10⁶ = 1000000.

About 400 years ago mathematicians had the idea that writing numbers as powers of 10 would make multiplication much easier. Instead of multiplying numbers together it would only be necessary to add the exponents. In the days before calculators anything that made calculations easier was very welcome.

A function that was called the logarithm was defined, written f(x) = lg x  or f(x) = log x where log a is the power that 10 has to be raised to in order to equal a.

The function f(x) = log x  is defined for all positive numbers and finds the power to which 10 has to be raised to give us x. We can only find the logarithm of positive numbers as powers of 10 are always poisitive.

The  Domain of the function f(x) = log x is therefore
{ xÎR | x > 0 }.

We will now look at some logarithms on a calculator. This function can be found on all Scientific calculators but we will use the Casio graphical calculator.

To find log 2 you do the following:

f(1) = log 1 = 0 because 10⁰ = 1

f(2) = log 2 ≈ 0.303 because  10^0.303 ≈ 2

f(3) = log 3 ≈ 0.477 because  10^0.477 ≈ 3

f(10) = log 10 = 1 because  10¹ = 10

f(20) = log 20 ≈ 1.303 because  10^1.303 ≈ 20

f(100) = log 100 = 2 because  10² = 100

f(1000) = log 1000 = 3 because  10³ = 1000

The function f(x) = log x is the inverse of the function g(x) = 10^x meaning that if one function is used on the outcome of the other then we get back the original number. This is the same idea as squaring a number and then taking it’s square root.

Example 1

We take the log of both sides of the equation.  As the functions log x and 10 x cancel each other out we are just left with x on the left hand side. Check this result on your calculator.

Example 2

We make both sides of the equation into powers of 10.  The functions log x and 10x cancel each other out and we are left with just x on the left side of the equation.

Logarithms are simply exponents ( indices ) or powers and therefore follow rules corresponding to to the rules for indices.

We know that 10^x×10^y = 10^x+y so it follows that if

x = log a and y = log b.

then log ab = log a + log b.

In the same we  see that log ^a/_b = log a − log b.

We can write  log aⁿ = log a + log a + ∙ ∙ ∙ ∙ + log a

(n terms) = n log a.

This gives us the following logarithm rules:

Example 3

The logarithm rules are used to combine the following into one logarithm term:

a)  log a + log b + log c = log abc

b)  log a + log 2b + log 3c + log 4 = log a×2b×3c×4 = log 24abc

c)  log a + log a + log a = 3 log a = log a³

d)  log 2a − log 2b = log 2 + log a − log 2 − log b = log ^a/_b

e)  ¼×(log a + log a³) = ¼×(log a + 3 log a) = ¼∙4 log a = log a

Example 4

Solve the equation by writing the left hand side of the equation as one logarithm.

a) 

b)

c)

d)

The correct solution is x = 5 because log(-5) does not exist.

The rule  log aⁿ = n log a  is used to solve equations of the type a^x = b, where the exponent is the unknown value.

We take the logarithm of both sides of the equation and then use this rule to move the x down.

    a^x = b

    log a^x = log b

    x log a = log b

Example 5 

Try Quiz 2 on Exponentials and logarithms.  

Remember to use the checklist to keep track of your work.