Problem 50 - A Graphical Approach to Precalculus with Limits

Question

Solve each logarithmic equation. Express all solutions in exact form. Support your solutions by using a calculator. $$2 \log _{2}(5 x-3)+1=17$$

Step-by-Step Solution

Answer

The exact solution is \(x = \frac{259}{5}\).

Step 1: Isolate the Logarithmic Expression

We start by isolating the logarithmic part of the equation. The given equation is: \[ 2 \log_{2}(5x - 3) + 1 = 17 \]Subtract 1 from both sides to get:\[ 2 \log_{2}(5x - 3) = 16 \]

Step 2: Divide to Solve for the Logarithm

To isolate \( \log_{2}(5x - 3) \), divide both sides of the equation by 2:\[ \log_{2}(5x - 3) = 8 \]

Step 3: Convert Logarithmic Equation to Exponential Form

The next step is to convert the logarithmic equation into an exponential form. Remember the definition of a logarithm, which can be written as:\[ a = \log_b(c) \Rightarrow b^a = c \]Thus, our equation becomes:\[ 5x - 3 = 2^8 \]

Step 4: Calculate the Exponent

Calculate \(2^8\):\[ 2^8 = 256 \]So the equation now is:\[ 5x - 3 = 256 \]

Step 5: Solve for x

Now, solve for \(x\). Add 3 to both sides:\[ 5x = 259 \]Then, divide both sides by 5:\[ x = \frac{259}{5} \]

Step 6: Exact Solution Verification

The solution expressed in exact form is \(x = \frac{259}{5}\). Use a calculator to confirm the solution:\[ x = 51.8 \]Thus, this confirms that our solution is accurate.

Key Concepts

Solving Logarithmic Equations

Logarithmic equations often may seem complex at first glance. However, they can be manageable if broken down into simple steps. Let's take the equation from the exercise: \(2 \log_{2}(5x - 3) + 1 = 17\).

The initial step in tackling this type of problem is isolating the logarithmic expression. You want to get the logarithmic term by itself on one side of the equation. This step is similar to handling algebraic equations, where you aim to isolate the variable. Here, we subtracted 1 from both sides yielding \(2 \log_{2}(5x - 3) = 16\).

Next, we need to isolate \(\log_{2}(5x - 3)\) completely by dividing every term by 2. Doing so simplifies the equation further to \(\log_{2}(5x - 3) = 8\).

Once the logarithmic part is isolated, it becomes much more feasible to solve.

Exponential Form

Understanding exponential form is crucial in solving logarithmic equations. When working with a logarithmic equation like \(\log_b(c) = a\), it is essential to know that this can be expressed in exponential form: \(b^a = c\). This transformation is a fundamental property of logarithms.

In the given exercise, once you have \(\log_{2}(5x - 3) = 8\), switching to exponential form involves recognizing \(5x - 3\) as \(c\), 2 as \(b\), and 8 as \(a\). Hence, rewriting the equation in exponential form gives us \(5x - 3 = 2^8\).

Converting to exponential form often simplifies the process because it turns a logarithmic problem into a straightforward power problem. When simplified, \(2^8 = 256\), resulting in \(5x - 3 = 256\), making the equation much easier to handle.

Exact Form Solution

An exact form solution is a detailed, precise answer rather than an estimated or approximate value. In this context, it means we solve \(5x - 3 = 256\) perfectly, without rounding.

First, add 3 to both sides of the equation, which simplifies to \(5x = 259\). It's crucial to maintain this exactness without moving to decimals too soon.

Afterwards, divide each side by 5 to find \(x\): \(x = \frac{259}{5}\). This fraction represents the exact form, providing the most accurate solution in terms of an equation context. For confirmation, you might translate this into a decimal using a calculator, yielding \(x = 51.8\).

This dual representation of the answer demonstrates both precision and practical understanding, showing how closely related exact form solutions and numerical approximations can be.