LOGARITHMS
(SCIENCE, TECHNOLOGY, ENGINEERING, AND MATHEMATICS)
WHAT ARE LOGARITHMS?
Logarithms, simply put, are exponents. The base of a logarithm is also called a base,
while the value when the base is raised to a logarithm is called the argument of the
logarithm.
WHY DO I NEED TO LEARN LOGARITHMS?
Logarithms make it possible to work with small numbers, even in dealing with actual
numbers that are in the millions, billions, or trillions. It allows us to look at numbers
in terms of the power of 10 that it involves.
WHAT DO LOGARITHMS LOOK LIKE?
HOW DO I WORK WITH LOGARITHMS?
The first skill needed is to convert an expression from its exponential form, to its
equivalent logarithmic form. This equivalence relation will allow us to isolate and
determine the base, the exponent, or the argument of the logarithm.
EXPONENTIAL FORM LOGARITHMIC FORM
bx=y log b y = x
“b raised to the power x is equal to y.” “The logarithm of y to the base b is x.”
The base in the exponential form is the subscript in its logarithmic form.
The exponent in the exponential form is what the logarithmic expression is equal
to.
The value of the exponential expression is equal to the argument of the logarithm.
1
, LOGARITHMS
(SCIENCE, TECHNOLOGY, ENGINEERING, AND MATHEMATICS)
3
Example: Convert 4 = 64 to logarithmic form
3
Solution: Using the equivalence relation shown above, 4 = 64 is equivalent to
𝑙𝑜𝑔464 = 3
1
Example: Convert 𝑙𝑜𝑔93 = 2
1/2
Solution: 9 = 3
1/2
(Note: 9 = square root of 9)
Example: Solve for k in 𝑙𝑜𝑔2𝑘 = 8
Solution: Convert the equation into exponential form.
8
2 = 𝑘
Evaluate the left-hand side of the equation.
8
k = 2 = 256
Any positive real number not equal to 1 can be the base of a logarithm. But there are
2 special kinds of logarithm.
The common logarithm is the logarithm to the base 10. This is denoted by omitting
the subscript in the logarithmic notation. That is log x is understood to be log 10 x. It
is referred to as “common” because it is commonly the logarithm that is
programmed in calculators and computers.
Examples:
2
𝑙𝑜𝑔 100 = 2, 𝑠𝑖𝑛𝑐𝑒 10 = 100
0
𝑙𝑜𝑔 1 = 0, 𝑠𝑖𝑛𝑐𝑒 10 = 1
2
(SCIENCE, TECHNOLOGY, ENGINEERING, AND MATHEMATICS)
WHAT ARE LOGARITHMS?
Logarithms, simply put, are exponents. The base of a logarithm is also called a base,
while the value when the base is raised to a logarithm is called the argument of the
logarithm.
WHY DO I NEED TO LEARN LOGARITHMS?
Logarithms make it possible to work with small numbers, even in dealing with actual
numbers that are in the millions, billions, or trillions. It allows us to look at numbers
in terms of the power of 10 that it involves.
WHAT DO LOGARITHMS LOOK LIKE?
HOW DO I WORK WITH LOGARITHMS?
The first skill needed is to convert an expression from its exponential form, to its
equivalent logarithmic form. This equivalence relation will allow us to isolate and
determine the base, the exponent, or the argument of the logarithm.
EXPONENTIAL FORM LOGARITHMIC FORM
bx=y log b y = x
“b raised to the power x is equal to y.” “The logarithm of y to the base b is x.”
The base in the exponential form is the subscript in its logarithmic form.
The exponent in the exponential form is what the logarithmic expression is equal
to.
The value of the exponential expression is equal to the argument of the logarithm.
1
, LOGARITHMS
(SCIENCE, TECHNOLOGY, ENGINEERING, AND MATHEMATICS)
3
Example: Convert 4 = 64 to logarithmic form
3
Solution: Using the equivalence relation shown above, 4 = 64 is equivalent to
𝑙𝑜𝑔464 = 3
1
Example: Convert 𝑙𝑜𝑔93 = 2
1/2
Solution: 9 = 3
1/2
(Note: 9 = square root of 9)
Example: Solve for k in 𝑙𝑜𝑔2𝑘 = 8
Solution: Convert the equation into exponential form.
8
2 = 𝑘
Evaluate the left-hand side of the equation.
8
k = 2 = 256
Any positive real number not equal to 1 can be the base of a logarithm. But there are
2 special kinds of logarithm.
The common logarithm is the logarithm to the base 10. This is denoted by omitting
the subscript in the logarithmic notation. That is log x is understood to be log 10 x. It
is referred to as “common” because it is commonly the logarithm that is
programmed in calculators and computers.
Examples:
2
𝑙𝑜𝑔 100 = 2, 𝑠𝑖𝑛𝑐𝑒 10 = 100
0
𝑙𝑜𝑔 1 = 0, 𝑠𝑖𝑛𝑐𝑒 10 = 1
2
