Write each equation in its equivalent exponential form. 2 = log₉ x

Verified step by step guidance

1. Recall the definition of logarithm: if \(y = \log_b x\), then the equivalent exponential form is \(b^y = x\).
2. Identify the base \(b\), the exponent \(y\), and the result \(x\) from the given equation \(2 = \log_9 x\).
3. Here, the base \(b\) is 9, the exponent \(y\) is 2, and the result \(x\) is the unknown.
4. Rewrite the logarithmic equation \(2 = \log_9 x\) in exponential form using the formula: \$9^2 = x$.
5. This gives the equivalent exponential equation, which expresses \(x\) in terms of the base and exponent.

Key Concepts

Definition of Logarithms

A logarithm answers the question: to what exponent must the base be raised to produce a given number? For example, log base 9 of x equals 2 means 9 raised to the power 2 equals x. Understanding this definition is essential to convert between logarithmic and exponential forms.

Conversion Between Logarithmic and Exponential Forms

Every logarithmic equation can be rewritten as an equivalent exponential equation. The general form log_b(a) = c is equivalent to b^c = a. This conversion is key to solving or rewriting logarithmic expressions in exponential terms.

Properties of Exponents

Exponents represent repeated multiplication of a base number. Knowing how to interpret and manipulate exponents, such as understanding that b^c means multiplying b by itself c times, helps in correctly expressing and simplifying exponential forms derived from logarithmic equations.