Problem 55 - Algebra and Trigonometry

Question

For what value of \(x\) is the following true? $$\log (x+3)=\log x+\log 3$$

Step-by-Step Solution

Answer

The value of \( x \) is \( \frac{3}{2} \).

Step 1: Understanding Logarithm Properties

We start by recalling that the property of logarithms states that \[\log a + \log b = \log(ab)\]. Using this property, we can rewrite the expression on the right-hand side. Therefore, we have: \[\log x + \log 3 = \log(3x)\].

Step 2: Equating Logarithmic Expressions

Now that both sides of the equation are logarithms, we use the fact that if \[\log a = \log b\], then \(a = b\). Thus, we write the equation as: \[x+3 = 3x\].

Step 3: Solving the Equation for x

Next, let's simplify and solve the equation\[x + 3 = 3x\]by subtracting \(x\) from both sides, leading to:\[3 = 2x\].Then, divide both sides by \(2\) to solve for \(x\):\[x = \frac{3}{2}\].

Step 4: Verification of Solution

Finally, verify the solution by substituting \(x = \frac{3}{2}\) back into the original logarithmic equation.Both sides should evaluate to the same value if the solution is correct. Calculate separately:Left-hand side: \[\log(\frac{3}{2} + 3) = \log(\frac{9}{2})\].Right-hand side: \[\log(\frac{3}{2}) + \log(3) = \log(\frac{9}{2})\]. Since both sides are equal, \(x = \frac{3}{2}\) is correct.

Key Concepts

Logarithmic Properties

Logarithms are powerful tools in mathematics that allow us to handle multiplicative relationships as additive ones. A fundamental property used in logarithms is the product rule. It states that the logarithm of a product is equal to the sum of logarithms of individual numbers. This means if you have \ \(\log a + \log b\ \), you can rewrite it as \ \(\log(ab)\ \). This property is crucial when simplifying expressions like \ \(\log x + \log 3\ \) into a single logarithm, \ \(\log(3x)\ \).
Understanding these properties is key to simplifying logarithmic equations and finding solutions effectively. They transform complicated equations into less complex ones, making calculations more manageable.

Solving Equations

Solving equations using logarithms often involves ensuring both sides of the equation match a specific form. In our example, we used the property \ \(\log a = \log b\ \) implies \ \(a = b\ \). This allows us to equate numbers directly without handling the logarithms at every step.
After using logarithmic properties to rewrite the original equation, \(\log(x+3) = \log(3x)\), it simplifies to \(x + 3 = 3x\). This type of transformation is where understanding underlying properties saves time and effort. The next step is simple arithmetic: simplifying and solving the equation for the variable \(x\), which shows we should subtract \(x\) from both sides to get \(3 = 2x\), and then divide by 2 to find \(x = \frac{3}{2}\).
Breaking down equations into simpler steps helps avoid mistakes and ensures an accurate solution.

Verification of Solutions

When you've found a solution to an equation, it's a good idea to verify it. Verification helps confirm correctness by substituting the solution back into the original equation to ensure both sides are equal. In our example, substituting \(x = \frac{3}{2}\) back, we evaluate both sides of the original logarithmic equation.

• Left-hand side: \ \log(\frac{3}{2} + 3) = \log(\frac{9}{2})\ \
• Right-hand side: \ \log(\frac{3}{2}) + \log(3) = \log(\frac{9}{2})\ \

Because both sides evaluate to the same value, the solution \(x = \frac{3}{2}\) is verified as correct.
Verification is a crucial step in problem-solving, ensuring that mistakes are caught and corrected before concluding the solution process.