Evaluate each expression using the change-of-base formula...

Evaluate each expression using the change-of-base formula and either base 10 or base $e$. Answer in exact form and in approximate form using nine decimal places, then verify the result using the original base.
$$\log _{0.2} 0.008$$

Key Concepts

Verification Using Exponential Functions

After calculating a logarithmic expression, verification can be done by converting the logarithmic equation back into its equivalent exponential form. This step confirms that raising the base to the calculated logarithm indeed yields the original number. It reinforces the understanding of the inverse nature of exponentiation and logarithms.

Exact vs Approximate Evaluation

When evaluating logarithmic expressions, it is often useful to provide an exact form—using algebraic expressions or known constants—and an approximate numerical value to a specified number of decimal places. Exact forms preserve mathematical accuracy, while approximate forms offer practical utility when precise numerical results are needed for further computations.

Logarithms

Logarithms are the inverse operations of exponentiation. They help determine the exponent to which a base must be raised to obtain a given number. Understanding logarithms is fundamental in solving equations where the unknown appears as an exponent and in modeling phenomena that exhibit exponential growth or decay.

Change-of-Base Formula

The change-of-base formula allows one to convert a logarithm from one base to another, typically to bases that are more convenient for computation, such as 10 or e. This formula states that log_base(a) can be expressed as the quotient of logarithms in any other base, i.e., log_base(a) = (log_newbase(a))/(log_newbase(base)). This is crucial for evaluating logarithms on calculators that may only support certain bases.

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