Write each equation in its equivalent logarithmic form. 2^-4 = 1/16

Verified step by step guidance

1. Identify the components of the exponential equation \(2^{-4} = \frac{1}{16}\). Here, the base is 2, the exponent is -4, and the result is \(\frac{1}{16}\).
2. Recall the definition of logarithms: If \(a^x = b\), then the equivalent logarithmic form is \(\log_{a} b = x\).
3. Apply this definition to the given equation by setting \(a = 2\), \(b = \frac{1}{16}\), and \(x = -4\).
4. Write the logarithmic form as \(\log_{2} \left( \frac{1}{16} \right) = -4\).
5. This expresses the original exponential equation in its equivalent logarithmic form.

Key Concepts

Exponential and Logarithmic Forms

An exponential equation like a^x = b can be rewritten in logarithmic form as log_a(b) = x. This conversion helps in solving for exponents by expressing the relationship between the base, exponent, and result in terms of logarithms.

Properties of Exponents

Understanding how exponents work, including negative exponents, is essential. For example, a negative exponent indicates the reciprocal of the base raised to the positive exponent, such as 2^-4 = 1/(2^4) = 1/16.

Logarithm Definition and Notation

A logarithm log_a(b) answers the question: 'To what power must the base a be raised to get b?' Recognizing this definition allows one to rewrite exponential equations into logarithmic form accurately.