Problem 39 - Algebra 2

Question

Solve each equation. Check your answers. $$ 3 \log x=1.5 $$

Step-by-Step Solution

Answer

The solution to the equation is \(x = \sqrt{10}\) or approximately 3.16228.

Step 1: Isolate the logarithm

The given equation is \(3 \log x = 1.5\). Divide both sides of the equation by 3 to isolate \(\log x\): \(\log x = 1.5 / 3 = 0.5\)

Step 2: Convert to exponential form

The logarithmic equation \(\log x = 0.5\) can be rewritten in exponential form as \(10^{0.5} = x\). The base of the logarithm (which is 10 in this case, as there's no base indicated in the equation) is raised to the value on the other side of the equation to become equal to x.

Step 3: Solve for x

Evaluating \(10^{0.5}\) we get \(x = \sqrt{10}\) or approximately 3.16228.

Step 4: Check the solution

Substitute \(x = \sqrt{10}\) back in the original equation \(3 \log x = 1.5\). We have \(3 \log \sqrt{10}\) which simplifies to \(1.5 = 1.5\). Thus, the solution is correct.

Key Concepts

Exponential Form

When solving logarithmic equations, converting them to exponential form can make finding solutions easier. In a logarithmic equation like \( \log x = 0.5 \), the base of the logarithm, without any explicit base written, is 10. To convert this equation into exponential form, you raise the base (10) to the power of the result equal to the argument, which is \( x \).

• This conversion results in the exponential equation \( 10^{0.5} = x \).
• An advantage of the exponential form is that it often makes it straightforward to calculate the value of \( x \).

By rewriting the equation, the process becomes simpler to manage, leading directly to the solution. It enables quicker calculations, helping you reach the final answer with less error potential.

Checking Solutions

It is crucial to verify whether the calculated solution for a logarithmic equation is correct. Checking the solution involves substituting the derived value back into the original equation to ensure equality holds.

• For example, substitute \( x = \sqrt{10} \) back into the original equation \( 3 \log x = 1.5 \).
• This involves calculating \( 3 \log \sqrt{10} \) and confirming it equals 1.5.

This check eliminates doubts about the accuracy, ensuring no errors occurred during the calculation process. It is a practical step to confirm your understanding and confidence in solving logarithmic problems effectively.

Logarithmic Isolation

The isolation of the logarithm in a given equation is a critical initial step. It involves manipulating the equation so that the logarithmic term stands alone on one side. In the example equation \( 3 \log x = 1.5 \), here is how you proceed:

• The goal is to make \( \log x \) isolated — to get it by itself before proceeding with further steps.
• Start by dividing both sides by 3, simplifying the equation to \( \log x = 0.5 \).

This simplification process allows a direct path to converting the equation into exponential form. Isolating the logarithm paves the way to uncovering \( x \) by reducing complexities, making the equation more manageable. It helps ensure clarity and step-by-step precision while working towards a solution.