In Exercises 101–104, write each equation in its equivalent exponential form. Then solve for x. log₄x=-3

Verified step by step guidance

1. Recall the definition of logarithm: if \( \log_b a = c \), then the equivalent exponential form is \( b^c = a \).
2. Apply this definition to the given equation \( \log_4 x = -3 \). Rewrite it as \( 4^{-3} = x \).
3. Express \( 4^{-3} \) as a fraction with a positive exponent: \( 4^{-3} = \frac{1}{4^3} \).
4. Calculate \( 4^3 \) by multiplying 4 by itself three times: \( 4 \times 4 \times 4 \).
5. Write the final expression for \( x \) as \( x = \frac{1}{4^3} \), which is the solution in exponential form.

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_b(a) = c means b^c = a. Understanding this definition is essential to convert between logarithmic and exponential forms.

Converting Logarithmic Equations to Exponential Form

To rewrite a logarithmic equation like log_b(x) = y in exponential form, express it as b^y = x. This conversion simplifies solving for the variable by changing the equation into a more straightforward exponential expression.

Solving Exponential Equations

Once the equation is in exponential form, solving for the variable involves evaluating powers or roots. For example, if 4^{-3} = x, calculate 4^{-3} by using the negative exponent rule, which means 1 divided by 4^3.