Exponents Logarithms functions are mutually inverse. So, they undo one another. That's it. A logarithm is an exponent in its simplest form. With the logarithm of one number, you find the exponent to be increased to generate the number again by a certain value known as a base. Let’s learn about both the topics in detail and also the conversion of the log to exponential form and also exponential to log form.
Exponents
Exponent is a power that raises a number, symbol or expression. The use of exponents is just a quick way to indicate that you want to multiply something many times by itself.
For example, let us say that we gave the following expression,
5 x 5 x 5
This can be written as 53 in exponential form.
They both are the same, that is 125, but writing in an exponent way is easier and shorter to write. This is useful when we have to multiply something a lot of times.
In the same example above, 53, 5 is referred to as the "base" and "3" is known as the "exponent". It is also called "5 to the power of 3". Therefore, sometimes exponent is called "the power of" number.
Let's see a couple of specific exponents:
Squared: We call it squared when something has 2 as an exponent.
Cubed: We called it cubed if some number has an exponent of 3.
Note: We must be cautious about an exponent of 0. Whenever an exponent of 0 is present, the answer is 1.
Logarithms
Logarithms are part of Mathematics. They are closely associated with exponential functions. A logarithm indicates what exponent (or power) a certain number requires in order to generate, and hence logarithms are the opposite of exponentiation.
Let’s look at the following example:
Here, we have an exponential function i.e., 23= 8
So, we have base as 2, the exponent as 3 and so the answer is 8.
We discussed that the logarithmic equation is the inverse of the exponential equation.
So, we can write it as Log2(8) = 3
In the above equation, the base is 2, the will be argument 8, and the answer is 3.
Log to Exponential
To convert logarithmic form to exponential form, identify the logarithmic equation's base and move the base to the other side to the equal sign. Moving the base will make the current number or variable into the exponent. Do not move anything but the base; the other numbers or variables will not change sides, and the word "log" will be dropped.
Example: Write the logarithmic equation y = log7 9 in the exponential form.
Ans:
y = log₇9 ⇒ 7\[^{y}\] = 9
↑ ↑
In this example, we are converting logarithm to exponential. So, the base 7 will be moved from the right side to the equal sign to the left side of the equal sign by turning y to the exponent.
Exponential to Log
To convert exponential form to logarithmic form, identify the base of the exponential equation and then move base to the other side of the equal sign and add the word “log”. Do not move anything but the base, the other numbers or variables will not change sides.
Example: Convert exponential equation 43 = 64 into the logarithmic form.
Ans:
4³ = 64 ⇒ 3 = log₄64
↑ ↑
Here, we can see that the base is 4, and the base moved from the left side of the exponential equation to the right side of the logarithmic equation, and the word “log” was added.
Log Exponent Rules
Log Rules:
logb(mn) = logb(m) + logb(n)
logb(m/n) = logb(m) – logb(n)
logb(mn) = n · logb(m)
The log rules could be expressed in less formal terms as:
Multiplication can be turned outside the log into addition and versa can be turned.
Division can be turned outside the log into a subtraction, and vice versa.
An exponent can be moved as a multiplier outwards on all within a log, and vice versa.
Exponent Rules:
When we multiply 2 terms by the same base, we can add both the exponents:
( xm ) ( xn ) = x( m + n )
When we have an exponent expression and that is raised to some power, you can simplify that by multiplying outer power to inner power:
( xm ) n = xm n
Anything to the power zero is just "1" (as long as that "anything" is itself not zero).
Conclusion
By using the rules of exponents, we can solve several exponential equations and rewrite each side with the same base as power. Then we use the fact that exponential functions are individually to compare the exponents and solve the unknown. If an exponential equation with a shared base cannot be rewritten, overcome by using each side's logarithm. In combination with skills learned in this like conversion of logarithmic form to exponential form, we can solve the equations which model real-world situations, whether an unknown is an exponent or an argument of a logarithm.
FAQs on Exponents Logarithms
1. What is the fundamental relationship between exponents and logarithms?
Exponents and logarithms are inverse operations. An exponential equation shows a base raised to a power to get a result (e.g., 23 = 8). The corresponding logarithmic equation shows what power the base needs to produce that same result (e.g., log2(8) = 3). Essentially, a logarithm solves for the exponent.
2. How do you convert an equation from exponential form to logarithmic form?
To convert an exponential equation like bx = y into its logarithmic form, you identify the base 'b'. The logarithm is the exponent 'x'. You write it as logb(y) = x. For example, the exponential equation 52 = 25 is converted to the logarithmic equation log5(25) = 2.
3. What are the key properties of logarithms used for solving problems?
The three most important properties of logarithms simplify complex expressions:
- Product Rule: The log of a product is the sum of the logs. logb(mn) = logb(m) + logb(n).
- Quotient Rule: The log of a quotient is the difference of the logs. logb(m/n) = logb(m) – logb(n).
- Power Rule: The log of a number raised to an exponent is the exponent times the log of the number. logb(mn) = n ⋅ logb(m).
4. Why are logarithms so important in mathematics and real-world applications?
Logarithms are crucial because they allow us to work with very large or very small numbers more easily. They transform multiplication and division into simpler addition and subtraction. In the real world, they are used to measure phenomena that have a huge range of values, such as earthquake intensity (Richter scale), sound levels (decibels), and the pH of chemical solutions.
5. How can you use logarithms to solve an equation where the variable is in the exponent?
To solve an equation like ax = b, you can take the logarithm of both sides. This allows you to use the power rule to bring the variable 'x' down from the exponent. The steps are:
- Take the log of both sides: log(ax) = log(b).
- Apply the power rule: x ⋅ log(a) = log(b).
- Isolate the variable: x = log(b) / log(a).
6. What is the difference between log(x + y) and log(x) + log(y)?
This is a common point of confusion. There is no simplification rule for log(x + y). The term must be evaluated as is. In contrast, log(x) + log(y) can be simplified using the product rule to become log(xy). It is a critical mistake to assume that log(x + y) is equal to log(x) + log(y).
7. What does it mean if the base of a logarithm is 'e'? What is a natural logarithm?
A natural logarithm is a logarithm with a special base called Euler's number, e (approximately 2.718). Instead of writing loge(x), it is abbreviated as ln(x). The natural logarithm is widely used in calculus, physics, and finance because it describes growth and decay processes that occur naturally, such as compound interest and population growth.
