Fill in the missing entries in each table. x 4 (1)/(2) log1 / 2(x)

step 1 So we're asked to solve for the missing entries. So let's start with our first column. So we're

Fill in the missing entries in each table. x ...

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Key Concepts

Definition of a Logarithm

A logarithm is the inverse operation of exponentiation, establishing the relationship between an exponent and its base. For any positive numbers a and b (with b ? 1), and any real number c, the logarithmic statement log_b(a) = c is equivalent to the exponential statement b^c = a. This definition forms the basis for solving equations involving logarithms and for converting between logarithmic and exponential forms.

Evaluating Logarithmic Functions

Evaluating a logarithmic function involves determining the exponent that a given base must be raised to in order to produce a specified number. This process typically includes rewriting the logarithmic expression in its exponential form, making it easier to solve for the unknown value. Such evaluations are essential when completing tables of values where either the input ('x') or the output (log_b x) is missing.

Properties of Logarithms with Bases between 0 and 1

When the base of a logarithm is a fraction between 0 and 1, the logarithmic function is decreasing, meaning that as the input increases, the logarithm’s value decreases. Understanding this property helps in anticipating the behavior of the function and verifying the consistency of tabulated values, ensuring that the relationship between the x-values and their logarithms is maintained correctly.

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