Introduction to natural log laws:
The natural logarithms are used for many features of real life, for example zooming, mirroring, rotating images, etc. The natural logarithms are defined by the exponent of the power to which a base number must be raised to equal a given number. The solving natural logarithms are also called the inverse of an exponents. In logarithm, it has two types of logarithms are the common logarithm, also called the base ten logarithms, and the natural logarithm, also called the base e logarithm. Let us see about natural logarithmic law in this article
The natural logarithms are written:
logex or ln x
Where e = 2.71828182846 (base of natural logarithm).
List of Natural Log Laws:
The following laws help in solving natural logarithms : [here, ln x = logex]
Product law:
` ln (x * y) = (ln x) + (ln y)` `or` `log_e (x * y) = (log_e x) + (log_e y)`
Quotient law:
`ln (x / y) = (ln x) - (ln y)` `or` `log_e (x / y) = (log_e x) - (log_e y)`
Power law:
`ln (x^n) = (n) ln x` `or` `log_e (x^n) = (n)log_e x` `Where` `^nsqrt(x) = x 1/n`
Reciprocal law:
`ln x = 1 / (log_xe)` `or` `log_e x = 1 / (log_xe)` `where` `x` ` represents` ` any` `Numbers.`
Examples for Solving Natural Logarithms by Using the Laws:
Example 1:
Rewrite the following natural logarithms by using the required laws.
`ln` `(a xx b^3 xx c)/(d^2)` .
Solution
Step 1: `ln` `(a xx b^3 xx c)/(d^2)` . [Given]
Step 2: `ln` `(a xx b^3 xx c) - ln(d^2)` . [Quotient laws]
Step 3: `ln` `(a) + ln b^3 + ln c - ln(d^2)` . [Product laws]
Step 4: `ln` `(a) + (3)ln (b) + ln (c) - 2ln(d)` . [Power laws]
This is the required rewritten natural logarithms.
Example 2
Rewrite the following natural logarithms by using the required laws.
`ln` `(x^2 xx sqrt(y))/(z^4)` .
Solution:
Step 1: `ln` `(x^2 xx sqrt(y))/(z^4)` . [Given]
Step 2: `ln` `(x^2 xx sqrt(y)) - ln(z^4)` . [Quotient laws]
Step 3: `ln` `(x^2) + ln(sqrt(y)) - ln(z^4)` . [Product laws]
Step 4: `ln` `(x^2) + ln(y^(1/2)) - ln(z^4)` . [Radical laws]
Step 5: `(2)ln` `(x) + (1/2)ln(y) - (4)ln(z)` . [Power laws]
This is the required rewritten natural logarithms.
Example 3: Solve the following natural logarithms by using the required natural log laws.
`ln 5^(7x-3) = 4` .
Solution:
Step 1: Given `ln 5^(7x-3) = 4`
Step 2: `(7x -3)ln 5 = 4` . [by using the power laws]
Step 3: `(7x -3) = 4/(ln 5)` . [by using the cross multiplications laws]
Step 4: Addition of 2 on both sides we get.
`(7x -3) +3 = 4/(ln 5) + 3` .
Step 5: `(7x) = (4 + 3ln(5))/(ln 5) ` . [By taking LCM]
Step 6: Now we are going to divide 7 on both sides we get.
`(7x)/7 = (4 + 3ln(5))/(7ln 5) ` .
Step 7: `(x) = (4 + 3ln(5))/(7ln 5) ` .
This is the required solved natural logarithms.
The natural logarithms are used for many features of real life, for example zooming, mirroring, rotating images, etc. The natural logarithms are defined by the exponent of the power to which a base number must be raised to equal a given number. The solving natural logarithms are also called the inverse of an exponents. In logarithm, it has two types of logarithms are the common logarithm, also called the base ten logarithms, and the natural logarithm, also called the base e logarithm. Let us see about natural logarithmic law in this article
The natural logarithms are written:
logex or ln x
Where e = 2.71828182846 (base of natural logarithm).
List of Natural Log Laws:
The following laws help in solving natural logarithms : [here, ln x = logex]
Product law:
` ln (x * y) = (ln x) + (ln y)` `or` `log_e (x * y) = (log_e x) + (log_e y)`
Quotient law:
`ln (x / y) = (ln x) - (ln y)` `or` `log_e (x / y) = (log_e x) - (log_e y)`
Power law:
`ln (x^n) = (n) ln x` `or` `log_e (x^n) = (n)log_e x` `Where` `^nsqrt(x) = x 1/n`
Reciprocal law:
`ln x = 1 / (log_xe)` `or` `log_e x = 1 / (log_xe)` `where` `x` ` represents` ` any` `Numbers.`
Examples for Solving Natural Logarithms by Using the Laws:
Example 1:
Rewrite the following natural logarithms by using the required laws.
`ln` `(a xx b^3 xx c)/(d^2)` .
Solution
Step 1: `ln` `(a xx b^3 xx c)/(d^2)` . [Given]
Step 2: `ln` `(a xx b^3 xx c) - ln(d^2)` . [Quotient laws]
Step 3: `ln` `(a) + ln b^3 + ln c - ln(d^2)` . [Product laws]
Step 4: `ln` `(a) + (3)ln (b) + ln (c) - 2ln(d)` . [Power laws]
This is the required rewritten natural logarithms.
Example 2
Rewrite the following natural logarithms by using the required laws.
`ln` `(x^2 xx sqrt(y))/(z^4)` .
Solution:
Step 1: `ln` `(x^2 xx sqrt(y))/(z^4)` . [Given]
Step 2: `ln` `(x^2 xx sqrt(y)) - ln(z^4)` . [Quotient laws]
Step 3: `ln` `(x^2) + ln(sqrt(y)) - ln(z^4)` . [Product laws]
Step 4: `ln` `(x^2) + ln(y^(1/2)) - ln(z^4)` . [Radical laws]
Step 5: `(2)ln` `(x) + (1/2)ln(y) - (4)ln(z)` . [Power laws]
This is the required rewritten natural logarithms.
Example 3: Solve the following natural logarithms by using the required natural log laws.
`ln 5^(7x-3) = 4` .
Solution:
Step 1: Given `ln 5^(7x-3) = 4`
Step 2: `(7x -3)ln 5 = 4` . [by using the power laws]
Step 3: `(7x -3) = 4/(ln 5)` . [by using the cross multiplications laws]
Step 4: Addition of 2 on both sides we get.
`(7x -3) +3 = 4/(ln 5) + 3` .
Step 5: `(7x) = (4 + 3ln(5))/(ln 5) ` . [By taking LCM]
Step 6: Now we are going to divide 7 on both sides we get.
`(7x)/7 = (4 + 3ln(5))/(7ln 5) ` .
Step 7: `(x) = (4 + 3ln(5))/(7ln 5) ` .
This is the required solved natural logarithms.
