Write each equation in its equivalent exponential form. 2 = log₃ 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 \(2 = \log_3 x\).
3. Here, the base \(b\) is 3, the exponent \(y\) is 2, and the result \(x\) is the unknown value inside the logarithm.
4. Rewrite the logarithmic equation \(2 = \log_3 x\) in exponential form using the formula: \$3^2 = x$.
5. This expresses the original logarithmic equation as an exponential equation, which is the equivalent form requested.

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 3 of x equals 2 means 3 raised to what power equals x. Understanding this definition is essential to convert between logarithmic and exponential forms.

Exponential Form of a Logarithmic Equation

The exponential form of a logarithmic equation log_b(x) = y is b^y = x. This equivalence allows rewriting logarithmic expressions as exponential ones, which is crucial for solving or simplifying equations involving logarithms.

Properties of Exponents

Exponents represent repeated multiplication of a base. Knowing how to interpret and manipulate expressions like b^y helps in understanding the relationship between logarithms and exponents, and in solving equations after conversion.