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Exponents and Logarithms: Laws, Definitions, and Inverses

This section covers the laws of exponents with a general base, fractional exponents, common logarithms, natural logarithms, and the laws of logarithms. It also explains how exponentials and logarithms are inverse functions.

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Exponents and Logarithms: Laws, Definitions, and Inverses

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  1. Section 4.2 Exponential and Logarithmic Functions

  2. LAWS OF EXPONENTS Laws of Exponents with General Base a: If the base number a is positive and x and y are any real numbers, then

  3. ADDITIONAL EXPONENT LAWS

  4. Recall that radicals can be expressed as fractional exponents. That is, Below are some examples. FRACTIONAL EXPONENTS

  5. LAWS OF EXPONENT WITH BASE e If x and y are real numbers, then

  6. COMMON LOGARITHMS Definition: The common logarithm of the positive number x is the power to which 10 must be raised in order to obtain the number x. It is denoted by log10 x. Thus, y = log10x means the 10y = x. Frequently, we omit the subscript 10 and simply write log x for the common logarithm of the positive number x.

  7. NATURAL LOGARITHMS Definition: The natural logarithm of the positive number x is the power to which e must be raised in order to obtain the number x. It is occasionally denoted by logex, but more frequently by ln x (with l for “log” and n for “natural”). Thus, y = ln x means that ey = x. NOTE: Only positive numbers have logarithms (common or natural).

  8. LAWS OF LOGARITHMS Laws of Logarithms: If x and y are positive real numbers, then • The logarithm of a product is the sum of the logarithms. • The logarithm of a quotient is the difference of the logarithms. • The logarithm of a reciprocal is the negative of the logarithm. • The logarithm of a power is the exponent times the logarithm of the base. • The logarithm of one is zero.

  9. EXPONENTS AND LOGARITHMS AS INVERSES Just as addition and subtraction (and multiplication and division) undo each other, exponentials and logarithms undo each other also. That is, eln x = x and ln ex = x. Two functions, that undo each other are called inverses.

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