EXPONENTS AND LOGARITHMS
Tutorial Manual
MUT
Maths 1
,Exponential and Logarithmic Functions
Learning Outcomes
After completing this section, you should be able to
Apply the laws of indices
Apply the laws of logarithms
Introduction
This section begins by reviewing the basic laws of indices and logarithms.
Understanding of these laws is crucial in all the other areas you will encounter
dealing with indices and logarithms, and that extends beyond this content area.
Let’s begin by reviewing these laws from what you already know from high
school.
Laws of Indices
The following laws of indices are true for all the non-zero values of a, m and n:
1. 𝑎𝑚 × 𝑎𝑛 = 𝑎𝑚+𝑛 eg 𝑎3 × 𝑎9 = 𝑎3+9 = 𝑎12
2. 𝑎𝑚 ÷ 𝑎𝑛 = 𝑎𝑚−𝑛 eg 𝑎2 ÷ 𝑎−5 = 𝑎2−(−5) = 𝑎2+5 = 𝑎7
3. (𝑎𝑚 )𝑛 = 𝑎𝑚𝑛 eg (𝑎3 )2 = 𝑎3×2 = 𝑎6
The above 3 laws are the basic laws of indices and from these, we can generate
other important results:
4. 𝑎0 = 1
1
5. 𝑎−𝑚 = 𝑎𝑚
1
6. 𝑎𝑚 = 𝑚√𝑎
1
, Exponential Functions
By definition, an exponential function is one where the unknown variable (x) is
in the index. Examples of exponential functions are 2𝑥 , 3−𝑥 and 10𝑥+1 .
Mathematically we say an exponential function is any function given by 𝑦 = 𝑎 𝑥 ,
where 𝑎 > 0 and a ≠ 1.
The diagram below shows some members of the exponential family:
NOTE:
All the curves pass through the point (1;0). This is because when 𝑥 = 0, 𝑦 =
𝑎 𝑥 = 𝑎0 = 1. Also, the x-axis is an asymptote for all exponential functions; that
is, the exponential functions approach the x-axis but don’t get to ‘touch’ or ‘cross’
it. We saw 𝑎 𝑥 > 0 for all values of 𝑥.
Base 𝒆
In this course and in all engineering calculations, the most important exponential
is the exponential function given to base 𝑒. The number 𝑒 is an irrational number
with many important uses in engineering and finance. Please note e is not an
abbreviation for exponential, it is just a symbol. You will get to appreciate its
importance as we move along with the cause. The value of 𝑒 from your calculators
is 2.718… This base will take over from base 10 that you were used to in high
school. The function 𝑦 = 𝑒 𝑥 has the same shape as the exponential functions
2
Tutorial Manual
MUT
Maths 1
,Exponential and Logarithmic Functions
Learning Outcomes
After completing this section, you should be able to
Apply the laws of indices
Apply the laws of logarithms
Introduction
This section begins by reviewing the basic laws of indices and logarithms.
Understanding of these laws is crucial in all the other areas you will encounter
dealing with indices and logarithms, and that extends beyond this content area.
Let’s begin by reviewing these laws from what you already know from high
school.
Laws of Indices
The following laws of indices are true for all the non-zero values of a, m and n:
1. 𝑎𝑚 × 𝑎𝑛 = 𝑎𝑚+𝑛 eg 𝑎3 × 𝑎9 = 𝑎3+9 = 𝑎12
2. 𝑎𝑚 ÷ 𝑎𝑛 = 𝑎𝑚−𝑛 eg 𝑎2 ÷ 𝑎−5 = 𝑎2−(−5) = 𝑎2+5 = 𝑎7
3. (𝑎𝑚 )𝑛 = 𝑎𝑚𝑛 eg (𝑎3 )2 = 𝑎3×2 = 𝑎6
The above 3 laws are the basic laws of indices and from these, we can generate
other important results:
4. 𝑎0 = 1
1
5. 𝑎−𝑚 = 𝑎𝑚
1
6. 𝑎𝑚 = 𝑚√𝑎
1
, Exponential Functions
By definition, an exponential function is one where the unknown variable (x) is
in the index. Examples of exponential functions are 2𝑥 , 3−𝑥 and 10𝑥+1 .
Mathematically we say an exponential function is any function given by 𝑦 = 𝑎 𝑥 ,
where 𝑎 > 0 and a ≠ 1.
The diagram below shows some members of the exponential family:
NOTE:
All the curves pass through the point (1;0). This is because when 𝑥 = 0, 𝑦 =
𝑎 𝑥 = 𝑎0 = 1. Also, the x-axis is an asymptote for all exponential functions; that
is, the exponential functions approach the x-axis but don’t get to ‘touch’ or ‘cross’
it. We saw 𝑎 𝑥 > 0 for all values of 𝑥.
Base 𝒆
In this course and in all engineering calculations, the most important exponential
is the exponential function given to base 𝑒. The number 𝑒 is an irrational number
with many important uses in engineering and finance. Please note e is not an
abbreviation for exponential, it is just a symbol. You will get to appreciate its
importance as we move along with the cause. The value of 𝑒 from your calculators
is 2.718… This base will take over from base 10 that you were used to in high
school. The function 𝑦 = 𝑒 𝑥 has the same shape as the exponential functions
2
