What does Logarithm mean?
A logarithm is a mathematical function that represents the exponent to which a fixed number (the base) must be raised to produce a given number. It is the inverse operation of exponentiation.
The expression logba = c means:
b is the base.
a is the number you are taking the logarithm of.
c is the exponent to which the base must be raised to get a.
Logarithms are comprised of parts from exponential equations in a way that a ^ x=b, then log (a) b =x
The rules of logarithms: Logarithms follow certain rules/laws that allows mathematicians to be able to solve them. There are a total of 7 which are as follows: Product, Quotient, Power, change of Base, zero property, identity property and inverse property.
Product Rule
The product rule states that when adding logarithms, you multiply the numbers of the logarithms. The same is done vise versa depending if you need to expand or condense logarithms.
Examples: 1) Log3(5)+Log3(7)=Log3(5*7)
2) Log5(2)+Log5(10)=Log5(2*10)
Quotient Rule
Much like the product rule, you take the numbers of logarithms being subtracted and you divide them instead of multiplying. Once again, it is vise versa depending on the goal you wish to achieve.
Examples: 1) Log6(5)-Log6(7)=Log6(5/7)
2) Log5(2)-Log5(3)=Log5(2/3)
Power Rule
The power rule is when you have to deal with logarithms having exponents. For this rule, you must take the exponent and multiply the logarithm by it. The same can be done if you have a root but instead of multiplying the logarithm, you divide. \
Examples
1)Log7(2²)= 2Log7(2)
2)Log2(5²)=2Log2(5)
Change Of Base
This rule is hardly used as much as before but it is still a rule that can be followed. It is used when you want to change the base of one logarithm to another. This method is usually done to change the logarithm base to either 10 or e.
examples:
1) Log6(10)= Log(10)/Log(6)
2) Log5(6)= Log(6)/Log(5)
Zero Property
Since logarithms are one way to write down exponential equations, this method abides by the same rules that come from exponents where if the logarithm number is 1 regardless of base, zero is the only power where any number is equal to 1.
Examples:
1) Log6(1)=0
2) Log5(1)=0
Identity Property
Identity property is similar to the zero property in the regard that it follows exponent rules. In this rule, if the base and the number of the logarithm are the same, the answer will be one since any number to the power of 1 is itself.
Examples:
1) Log4(4)=1
2) Log10(10)=1
Inverse Property
This one can be one of the more complicated ones to explain but in short, logarithms follow the exponential equations rules and so if the logarithm (contradicts) itself, it cancels out leaving you with the answer.
Examples:
1) 10^log10(7)=7
2) Log2(2⁴)=4
Reflection
Like everything else in math/algebra, it follows rules that can be either easy to understand or can cause confusion to some. However, we must always persevere and attempt to understand how each rule is used and how to identify them. Logarithms can indeed be hard especially with subtopics like exponential and logarithmic equations.