Problem 49 - College Algebra

Question

Solve each logarithmic equation in Exercises \(49-92 .\) Be sure to reject any value of \(x\) that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$ \log _{3} x=4 $$

Step-by-Step Solution

Answer

The exact solution to the logarithmic equation \( \log _{3} x = 4 \) is \( x = 81 \).

Step 1: Converting Logarithmic form to Exponential form

The logarithmic equation \( \log _{3} x = 4 \) can be written in exponential form according to the formula \( log_b(a) = c \) equivalently represented as \( b^c = a \). Hence \( 3^4 = x \).

Step 2: Evaluating the Exponential Expression

To evaluate this exponential equation, simply calculate \( 3^4 \), which equals 81.

Step 3: Checking the Solution

Check whether the solution, \( x = 81 \), is in the domain of the original logarithmic expression \( \log _{3} x \). Since the base of a logarithm can only be a positive number, not equal to 1, and the number inside the logarithm (the argument) must be positive, the solution satisfies all these conditions. Therefore, \( x = 81 \) is a valid solution.

Key Concepts

Converting Logarithmic to Exponential Form

Understanding how to convert logarithmic equations to exponential form is crucial for solving logarithmic problems effectively. Let's look at a simple example: If you have the logarithmic equation \( \log_3 x = 4 \) it represents the power to which you must raise 3 to obtain x. The exponential form of this equation is \( 3^4 = x \) which makes it clearer what x actually is.In general, the conversion follows this pattern: if \( \log_b(a) = c \) then \( b^c = a \) where b is the base of the logarithm, a is the result you get when raising b to the power of c, and c is the logarithm's value. Converting to exponential form is one of the first and crucial steps in solving logarithmic equations, and it allows us to proceed to find the numerical value of x.

Evaluating Exponential Expressions

Once we have the logarithmic equation in exponential form, the next step is evaluating the exponential expression. With our example \( 3^4 = x \), we simply calculate the exponent to find x. Here, \( 3^4 \) equals 81, so x must be 81. Evaluating exponential expressions requires understanding of exponents and sometimes a calculator for larger or more complex bases and exponents. Crucially, this step can be done by hand or with technology, depending on the complexity, but the key is knowing that the exponential expression represents the repeated multiplication of the base.

Domain of Logarithmic Functions

The domain of a function represents all the possible input values (x-values) that will yield a valid output without causing the function to become undefined or result in a math error. For logarithmic functions, this domain includes only positive real numbers. Why? Because you can only take the logarithm of a positive number. This is due to the fact that the logarithmic function represents the inverse of exponentiation and because there's no real number that can be raised to any power to get a negative number or zero. When solving logarithmic equations, it's essential to ensure that the solutions fall within the domain of the original logarithm. As in our example, \( x = 81 \) is positive and thus within the domain of the logarithmic function \( \log_3(x) \) ensuring that our solution is valid.