Simplify the expressions. a. 5^log5 7 b. 8^log8 √(2) c. 1.3^log1.3 75 d. log4 16 e. log3 √(3) f.

step 1 Our goal for this problem is to simplify log expressions. So we have, for example, 5 to the log

Simplify the expressions. a. 5^log5 7 b. 8^log8 √(2) c....

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Key Concepts

Representation of Radicals as Fractional Exponents

Understanding that radicals can be expressed as fractional exponents (for example, the square root of a number can be written as the number raised to the power of one-half) is a key concept. This conversion allows the simplification of logarithmic expressions involving roots by utilizing the properties of exponents and logarithms.

Logarithm of the Reciprocal

This concept involves the property that the logarithm of a reciprocal of the base is the negative of one, as in log_a(1/a) = -1. Recognizing and applying this property is important for simplifying logarithmic expressions that include reciprocal or inverse values.

Exponent-Logarithm Inversion

This concept refers to the property that an exponent with a logarithmic exponent of the same base simplifies directly to the argument of the logarithm. In general, a^(log_a(b)) = b. This property is essential for simplifying expressions where the base of the exponent and the logarithm match, as seen in many of the problems.

Logarithm of a Power

This concept involves the property that the logarithm of a power of the base simplifies to the exponent itself, i.e., log_a(a^x) = x. This idea is central when dealing with logarithmic expressions where the input is an exponentiation that can be directly matched to the base.

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