Write each equation in its equivalent logarithmic form. b³ = 1000

Verified step by step guidance

1. Identify the components of the exponential equation \(b^3 = 1000\). Here, \(b\) is the base, \$3\( is the exponent, and \)1000$ is the result.
2. Recall the definition of logarithms: If \(a^x = y\), then the equivalent logarithmic form is \(\log_a y = x\).
3. Apply this definition to the given equation by setting the base \(a\) as \(b\), the result \(y\) as \$1000\(, and the exponent \)x\( as \)3$.
4. Write the logarithmic form as \(\log_b 1000 = 3\).
5. This expresses the original exponential equation in its equivalent logarithmic form.

Key Concepts

Exponential and Logarithmic Forms

Exponential and logarithmic forms are two ways to express the same relationship. An equation like b^3 = 1000 can be rewritten in logarithmic form as log base b of 1000 equals 3. This conversion helps solve for unknown exponents or bases.

Definition of a Logarithm

A logarithm answers the question: to what power must the base be raised to get a certain number? Formally, if b^x = y, then log base b of y equals x. Understanding this definition is essential for rewriting exponential equations as logarithms.

Properties of Exponents

Properties of exponents, such as b^m * b^n = b^(m+n), help manipulate and simplify exponential expressions. Recognizing these properties aids in understanding the structure of the equation before converting it to logarithmic form.