Problem 103 - College Algebra

Question

Write each equation in its equivalent exponential form. Then solve for \(x .\) $$ \log _{4} x=-3 $$

Step-by-Step Solution

Answer

\(x = 1/64\)

Step 1: Convert to Exponential Form

The logarithmic equation \(\log _{4} x=-3 \) is equivalent to the exponential equation \(4^{-3} = x\) according to the logarithmic to exponential conversion rule \(\log _{b} a = c\) is equivalent to \( b^c = a\) . So, We have \(4^{-3} = x\)

Step 2: Solve for \(x\)

Next step is to solve for \(x\). We can start by calculating \(4^{-3}\) which is 1 divided by \(4^3\) or \(1/64\). Thus, \(x = 1/64\)

Key Concepts

Solving Logarithmic Equations

To solve logarithmic equations, it's essential to understand the relationship between logarithms and exponents. A logarithmic equation involves a logarithm of a variable, and to solve for that variable, one must often first convert the logarithmic equation to its equivalent exponential form. For example, the logarithm equation \(\log_b x = y\) can be rewritten as \(b^y = x\), where \(b\) is the base of the logarithm, \(x\) is the result, and \(y\) denotes the exponent.

Once converted, the exponential equation may be simpler to solve, as it becomes a matter of basic algebra. In the case of \(\log_{4} x = -3\), by following these steps, we identify that \(x\) must be \(\frac{1}{64}\), because \(\frac{1}{64}\) is the value that, when raised to the power of -3, equals \(\frac{1}{4^3}\). This process of reconceiving logarithmic terms in their exponential counterparts is a fundamental tool in solving logarithmic equations.

Exponential Equations

Exponential equations are equations where the unknown variable appears as an exponent. They are solved by finding a value for the variable that makes the equation true. Many times, these types of equations require the logarithmic functions to isolate the variable. However, in some cases, as with our example of converting \(\log_{4} x = -3\) to \({4^{-3} = x}\), the conversion simplifies the expression, leveraging the properties of exponential functions.

The key to solving exponential equations is often finding a common base between the terms of the equation or using logarithms to bring down the exponent. It's important to be comfortable with the laws of exponents, as these rules apply when manipulating exponential expressions to solve for variables. In cases where conversion is straightforward, such as the step-by-step solution provided, the calculation reduces to simple arithmetic.

Logarithms

Logarithms are the inverse operations of exponentiation. They answer the question, 'To what exponent must the base be raised, to produce a given number?'. For instance, \(\log_b a = c\) indicates that \(b^c = a\), with \(b\) being the base, \(a\) the number we want to find the logarithm of, and \(c\) the actual logarithm answer.

There are several properties of logarithms which are useful in various mathematical applications. These include the product, quotient, and power rules, which help in simplifying complex logarithmic expressions. Understanding the concept of logarithms, their properties, and how they relate to exponentiation is crucial, as it opens the way to solving many algebraic problems involving exponential growth, decay, and compound interest calculations.