LOGARITHMS

Short Questions and Answers on Logarithms

Q1: What is a logarithm?

Ans1: A logarithm reveals the exponent you must raise a base to in order to get a specific number.

Q2: What defines common logarithms?

Ans2: Common logarithms operate on a base of 10. Consequently, if log10​ x=y, we write it concisely as log(x)=y.

Q3: How are logarithmic and exponential functions related?

Ans3: Logarithmic and exponential functions are fundamentally inverse operations. For example, if we have the exponential equation b^y=x, we can write its inverse as the logarithmic equation logb ​x=y.

Q4: Can you describe the natural logarithm?

Ans4: The natural logarithm uses the base e, also known as Euler’s number (approximately 2.718). We denote this function as ln(x).

Q5: How do you simplify the expression log(ab)?

Ans5: We simplify the logarithm of a product using the product rule, which allows us to rewrite log(ab) as the sum of two logarithms: log(a)+log(b).

Q6: What is the change of base formula for logarithms?

Ans6: The change of base formula enables you to convert a logarithm from one base to another. The formula is logb​a=logc​ b logc ​a​, where ‘c’ can be any chosen base.

Q7: How do you simplify the expression log(a/b)?

Ans7: We use the quotient rule to simplify the logarithm of a quotient. This rule states that log(a/b)=log(a)−log(b).

Q8: Which logarithmic property allows you to move exponents to the front?

Ans8: The power rule gives us this ability. It allows you to move the exponent of a logarithmic term to the front as a coefficient, as shown by the formula n logb​ x=logb​x^n.

Q9: Solve for ‘x’ in the equation log2^​x=3.

Ans9: To solve this, simply convert the logarithmic equation to its exponential form. Therefore, x=2^3, which gives us a solution of x=8.

Q10: How can you express log(x) using ln(x)?

A10: We can use the change of base formula to show their relationship. This results in the expression log(x)=ln(10) ln(x)​.

FAQ: Frequently Asked Questions

Q1: Can we define the logarithm of a negative number?

Ans: For real numbers, the logarithm of a negative number is undefined. However, it involves complex numbers if you consider imaginary logarithms.

Q2: What is the relationship between logarithms and exponents?

Ans: Logarithms and exponents are inverse operations. For example, if b^y=x, then we can write that logb​  x=y, and vice versa.

Q3: How do you solve logarithmic equations?

Ans: To solve a logarithmic equation like logb​ x=y, you should rewrite it in exponential form as b^y=x and then solve for the variable.

Q4: What is the logarithmic identity for division?

Ans: The logarithmic identity for division is log(a/b)=log(a)−log(b).