Problem 39 - Algebra and Trigonometry

Question

Solve the logarithmic equation for \(x\) $$ \log (3 x+5)=2 $$

Step-by-Step Solution

Answer

The solution to the equation \( \log(3x+5)=2 \) is approximately \( x \approx 31.67 \).

Step 1: Understand the Logarithmic Equation

The given equation is \( \log (3x+5) = 2 \). This means we are looking for the value of \( x \) that makes this equality true.

Step 2: Convert the Logarithm Equation to Exponential Form

To solve \( \log (3x+5) = 2 \), convert the logarithmic equation to its equivalent exponential form. Recall that if \( \log_b(a) = c \), then \( a = b^c \). Here, the base is 10 (common logarithm), so the equation becomes: \[ 3x+5 = 10^2 \].

Step 3: Solve the Exponential Equation

Now, solve the equation \( 3x + 5 = 100 \). Start by isolating the term with \( x \):Subtract 5 from both sides:\[ 3x = 95 \].

Step 4: Solve for x

Divide both sides by 3 to solve for \( x \):\[ x = \frac{95}{3} \].

Step 5: Simplify the Solution

Simplify the fraction:\[ x = 31.67 \] (approximately, as a decimal).

Key Concepts

Exponential Form

When dealing with logarithms, converting the equation into an exponential form simplifies the process of finding the solution. A logarithmic equation like \( \log_b(a) = c \) can be transformed by remembering that it implies \( a = b^c \). For example, in the exercise \( \log(3x+5) = 2 \), the implicit base is 10, frequently used in common logarithms. Therefore, the equation converts to \( 3x+5 = 10^2 \), which equals 100.

• This transformation is helpful because it restates the problem in a straightforward algebraic equation that can be easily solved.
• It allows us to utilize properties of exponents, making the problem more manageable and less abstract for beginners.

Remember that mastering the transition from a logarithm to its exponential form is crucial. It's a fundamental part of solving logarithmic equations, providing a clearer path to determine unknown variables.

Solving Equations

Once the logarithmic equation is converted to its exponential form, solving it becomes a task of basic algebra. For the equation \( 3x+5=100 \), follow these steps to solve for \( x \):

• First, isolate the term \( 3x \) by subtracting 5 from both sides. This gives \( 3x = 95 \).
• Next, divide each side by 3, simplifying around the division: \( x = \frac{95}{3} \).
• The solution is \( x = 31.67 \) when rounded to two decimal places, but you could leave it in fraction form for exactness.

By following these steps:

• You are applying fundamental algebraic skills: isolating variables and balancing equations.
• Understanding the balance needed in the equation helps in developing math proficiency that is useful across a range of problems.

Learning the solving process is essential for tackling more complicated equations involving different operations and functions.

Common Logarithm

A common logarithm specifically refers to logarithms with base 10. They are usually denoted by \( \log \) without explicitly writing the base, as seen in the equation \( \log(3x+5)=2 \). Such logarithms are widely used because they simplify calculations and are aligned with the decimal system.

• Understanding that the base of a common logarithm is 10 helps in converting it effectively to exponential form. This is because powers of 10 are easy to compute mentally (like \( 10^2 = 100 \)).
• Common logarithms are extensively used in various scientific fields and real-world applications, from measuring sound intensity (decibels) to calculating pH in chemistry.

discern understanding of common logarithms is crucial:

• They provide a natural stepping stone to learning other logarithmic bases and applications.
• Being comfortable with base 10 logs serves as a useful reference point when learning more advanced logarithmic concepts.

Ultimately, this foundational knowledge aids in interpreting and solving problems swiftly and accurately across academic and practical applications.