Indices and Logarithms Quiz

Indices and Logarithms

In mathematics, indices and logarithms are closely related concepts. While indices involve raising a number to a certain power, logarithms represent the inverse operation. In particular, if a is a non-zero real number, the logarithm of a to base b, often denoted as log_ba, is the solution to the equation b^k=a. For example, log_10(1000) = 3 ("the log base 10 of 1000 is 3").

Logarithmic Functions

A logarithmic function is typically represented as f(x)=logbx, where b is the base. The domain of such a function is (0,∞) if b>1 or (-∞,0) if b<1. The range is (−∞,∞) regardless of the choice of b>1.

Laws of Logarithms

Product Rule

The product rule states that for positive numbers a and b, the logarithm of their product is the sum of the logarithms of a and b, i.e., log_ba+log_bc=log_bd. For example, if we have two numbers x and y, then ln(xy)=ln(x)+ln(y). This rule can be extended to any number of factors, as long as they are positive.

Quotient Rule

The quotient rule states that for positive numbers a and b, the logarithm of their quotient is the difference between the logarithms of a and b, i.e., log_ba−log_bc=log_bd. For example, if we have two numbers x and y, then ln(x/y)=ln(x)−ln(y). This rule also holds true for any number of terms, provided they are positive.

Power Rule

The power rule states that for a positive number a and a real exponent k, we have log_ba=klna. For example, if we have a number x raised to the power of n, then ln(x^n)=nln(x). This rule also holds true for any base.

Log of One and Log Reciprocal Rule

The log of one is always zero, regardless of the base. That is, log_be=0. Also, the log of the reciprocal of a is equal to the negative of the log of a, i.e., log_ba=−log_ba.

Common Log

Logarithms with base 10 are often referred to as common logs or decimal logs. If we write log_10x=a, it means that 10^a=x. In other words, x can be expressed as the product of powers of 10. For instance, log_10(1000)=3 because 10^3=1000. It's worth noting that the inverse operation, raising 10 to a power, is expressed as 10^a=x, where x is the original number.

Natural Log

Natural logs use Euler's constant e as the base, which is approximately equal to 2.718281828459... The natural logarithm of a number x is denoted as ln(x), and it satisfies the property that e^(ln(x))=x. Note that while natural logs do not explicitly show the base e, they implicitly assume this base when no other base is specified. For example, ln(e^x)=x because e^ln(e^x)=(e^x)^(1)=e^x=e^x.