In Exercises 1–8, write each equation in its equivalent exponential form. 2 = log3 x

Verified step by step guidance

1. Understand that the given equation is in logarithmic form: \(2 = \log_3 x\).
2. Recall the definition of a logarithm: \(\log_b a = c\) means \(b^c = a\).
3. Identify the base \(b\), the exponent \(c\), and the result \(a\) from the logarithmic equation: here, \(b = 3\), \(c = 2\), and \(a = x\).
4. Rewrite the equation in exponential form using the identified components: \(3^2 = x\).
5. Verify the conversion by checking that the exponential form correctly represents the original logarithmic equation.

Key Concepts

Logarithmic Functions

Logarithmic functions are the inverses of exponential functions. The logarithm log_b(a) answers the question: 'To what exponent must the base b be raised to produce a?' In the given equation, log3(x) indicates that 3 must be raised to a certain power to yield x.

Exponential Form

Exponential form expresses equations in terms of exponents. For a logarithmic equation like log_b(a) = c, the equivalent exponential form is b^c = a. This transformation is crucial for solving equations involving logarithms, as it allows for easier manipulation and understanding of the relationship between the variables.

Base of a Logarithm

The base of a logarithm is the number that is raised to a power to obtain a given value. In the equation log3(x), the base is 3. Understanding the base is essential for converting logarithmic equations to exponential form, as it determines the relationship between the logarithm and the resulting exponential expression.