Problem 103 - Algebra and Trigonometry

Question

In Exercises 101–104, write each equation in its equivalent exponential form. Then solve for x. $$ \log _{4} x=-3 $$

Step-by-Step Solution

Answer

The solution is \( x = \frac{1}{64} \)

Step 1: Convert logarithmic form to exponential form

The equation can be converted to exponential form using the formula \( \log_b{a} = n \) convert to base exponential form is \( b^n = a \). Substituting \( b = 4 \), \( a = x \), and \( n = -3 \), we get \( 4^{-3} = x \)

Step 2: Evaluate the exponential expression

The expression \( 4^{-3} \) can be simplified to \( \frac{1}{4^3} \) or \( \frac{1}{64} \). This is because any number raised to a negative exponent is the reciprocal of the number raised to the positive exponent.

Step 3: Solve for x

Equating the expression in exponential form to \( x \), implies \( x = \frac{1}{64} \)

Key Concepts

Converting Logarithms to Exponentials

Understanding the relationship between logarithms and exponentials is essential for solving equations that involve logarithms. When you encounter a logarithmic equation like \( \log_{b} a = n \), it’s often easier to address it by converting it to its equivalent exponential form, which is expressed as \( b^{n} = a \). This conversion is crucial because it transforms the problem into an exponential equation that is generally more straightforward to solve.

For example, if you have the logarithmic equation \( \log_{4} x = -3 \), apply this conversion to get the exponential form, which results in \( 4^{-3} = x \). This step takes you from a logarithmic expression to one involving an exponent, making it easier to evaluate and solve for the variable.

Solving for x in Exponential Equations

Once you've converted a logarithmic expression to an exponential equation, the next task is to isolate and solve for the variable, typically represented as 'x'. In exponential equations like \( a^{n} = x \), 'a' is the base, 'n' is the exponent, and you're trying to find the value of 'x' that makes the equation true.

To solve these equations, it's often helpful to rewrite the exponential expression in a way that is easier to calculate. Sometimes, this involves breaking down the base into simpler components or making the exponent positive if it's negative. After rewriting, you'll arrive at a result for 'x' that satisfies the original equation.

Negative Exponents

Negative exponents can seem intimidating at first, but they follow a simple rule that helps demystify them. An expression with a negative exponent, like \( a^{-n} \), means that you take the reciprocal of the base raised to the positive exponent.

Specifically, \( a^{-n} \) is equal to \( \frac{1}{a^{n}} \). This also implies that when you have an equation with a negative exponent, converting the negative exponent to a positive one will often make the equation simpler to understand and solve. For example, when you see \( 4^{-3} \), remember that it's equivalent to \( \frac{1}{4^{3}} \), which is easier to compute.

Exponential Expressions

An exponential expression is characterized by a base raised to a power, indicating how many times the base is multiplied by itself. The general form of an exponential expression is \( a^{n} \), where 'a' is the base, and 'n' is the exponent or power.

The properties of exponents play a pivotal role in simplifying and solving exponential expressions. When dealing with these expressions, such as \( 4^{-3} \) which simplifies to \( \frac{1}{4^{3}} \) or \( \frac{1}{64} \), remember that evaluating them correctly will lead you to the solution of 'x'. By mastering the rules that govern exponents, you can tackle a wide variety of problems involving exponential growth or decay, compound interest, and more mathematical applications.