Problem 93 - College Algebra with Modeling and Visualization

Question

Solve each equation. Approximate answers to four decimal places when appropriate. $$4 \log _{2} x=16$$

Step-by-Step Solution

Answer

The solution is \(x = 16\).

Step 1: Isolate the logarithmic expression

Start with the equation \(4 \log_{2} x = 16\). Divide both sides by 4 to isolate the logarithmic expression. This gives \(\log_{2} x = 4\).

Step 2: Convert the logarithmic equation to an exponential equation

Recall the definition of a logarithm: if \(\log_{b} a = c\), then \(a = b^c\). Applying this to \(\log_{2} x = 4\), convert it to \(x = 2^4\).

Step 3: Solve for \(x\)

Calculate the value of \(x = 2^4\), which equals 16.

Key Concepts

Logarithmic Properties

Logarithms have unique properties that make them a powerful tool in solving equations. A logarithm essentially tells us how many times we need to multiply a number (the base) by itself to get another number. When you see an expression like \(\log_{b} a\), it means "the power you need to raise \(b\) to produce \(a\)." Several properties can simplify complex logarithmic equations, including:

• Product Property: \(\log_{b}(mn) = \log_{b}(m) + \log_{b}(n)\)
• Quotient Property: \(\log_{b}\left(\frac{m}{n}\right) = \log_{b}(m) - \log_{b}(n)\)
• Power Property: \(\log_{b}(m^n) = n \cdot \log_{b}(m)\)

In the original exercise, we used the Power Property to simplify the equation. We had \(4 \cdot \log_{2} x\), which is already in simplified form. Understanding these properties allows us to manipulate and solve logarithmic equations efficiently. When you isolate the logarithm, as in Step 1 of our exercise, the goal is to diagnose the equation for further simplification and conversion.

Exponential Equations

Once you isolate the logarithmic expression in an equation, the next step is often converting it to an exponential equation. The definition of a logarithm is crucial here: if you have \(\log_{b} a = c\), then it can be rewritten in exponential form as \(a = b^c\). This conversion makes the numbers work in a familiar way. In the exercise, after isolating \(\log_{2} x = 4\), we converted it to \(x = 2^4\). This step translates the logarithm into a straightforward multiplication problem. Exponential equations are what we call numbers raised to a power, and they often appear simpler than logarithmic expressions due to their direct computation.Recognizing and converting to exponential equations can greatly simplify the process of solving logarithmic equations. Plus, it builds an intuitive understanding of the logarithm-exponential relationship, making future problems easier to tackle.

Solving Equations

Solving equations accurately often requires systematic steps to isolate the variable of interest. In this exercise, that process began by dividing both sides of the equation by 4 to isolate the logarithmic part. Each time you perform an operation, you're working towards a clearer view of the variable.Once converted into an exponential form, as with \(x = 2^4\), solving becomes a matter of calculating the resulting expression. Here, \(2^4\) means multiplying 2 by itself four times, which results in 16.When solving equations, remember:

• Check your work: Plug the solution back into the original equation to verify correctness.
• Approximate as needed: Often, solutions might require rounding. Always follow given instructions on precision.

In summary, solving equations involves isolating the variable, using properties wisely, and verifying every step. This method ensures you're working accurately, and deeply understanding each part of the equation makes complex calculations more straightforward.