Problem 41 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

\(x = \frac{95}{3} \) or \( x = 31.6667 \)

Step 1: Understand the Logarithmic Equation

The equation given is \( \log (3x + 5) = 2 \). This means that \( 3x + 5 \) is the result of raising the base of the logarithm to the power of 2. In general, \( y = \log_b(z) \) translates to \( z = b^y \), where \(b\) is the base of the logarithm. Here, it is an implied base 10 logarithm (common logarithm).

Step 2: Convert the Logarithmic Equation to Exponential Form

Using the property of logarithms: \( \log_b(z) = y \Rightarrow z = b^y \). The equation becomes:\[ 3x + 5 = 10^2 \] because the base \( b \) of the logarithm is 10 and the value \( y \) is 2.

Step 3: Simplify the Exponential Equation

Calculate \( 10^2 \), which equals 100. Substitute this into the equation:\[ 3x + 5 = 100 \].

Step 4: Solve for \( x \)

Subtract 5 from both sides of the equation to isolate the term with \( x \):\[ 3x = 100 - 5 \]\[ 3x = 95 \].Now divide both sides by 3:\[ x = \frac{95}{3} \].

Step 5: Calculate the Division

Divide 95 by 3 to get the value of \( x \):\[ x = 31.6667 \] or as a fraction \( x = \frac{95}{3} \).

Key Concepts

Exponential Form

The exponential form is a way of expressing numbers that have been multiplied by themselves a certain number of times. It's an essential concept in mathematics, particularly when dealing with logarithms. In our exercise, the logarithmic equation \( \log (3x + 5) = 2 \) is converted to its exponential form to solve for \( x \).
The general rule for converting a logarithm to its exponential form is \( y = \log_b(z) \Rightarrow z = b^y \). This means the logarithmic base "\( b \)" raised to the power of "\( y \)" equals "\( z \)".
In our scenario, since we are dealing with common logarithms where the base \( b \) is 10, the equation transforms to \( 3x + 5 = 10^2 \). Here, the process involves understanding that the base 10 raised to the power of 2 will give the result of the expression inside the logarithm, which is \( 3x + 5 \). This transition from logs to exponentials is crucial to simplifying and solving the equation.

Common Logarithm

A common logarithm is a logarithm with a base of 10, often written simply as \( \log(x) \) without explicitly mentioning the base. It's widely used due to its practical application in sciences and engineering.
In the exercise at hand, the common logarithm is employed in the equation \( \log(3x + 5) = 2 \). Understanding that the base is 10 makes it easier to convert the logarithmic form into the exponential form, as we did with \( 3x + 5 = 10^2 \). Here, "2" is the exponent, indicating how many times you'll multiply 10 by itself to reach the quantity \( 3x + 5 \).
Recognizing when and how to apply this concept is invaluable for solving equations involving logarithms. Often, calculators use common logarithms to simplify computations due to the base 10, and students should be familiar with this when interpreting log expressions.

Solving Equations

Solving equations is all about finding the value of the unknown variable. After converting the logarithmic equation to exponential form, we arrive at the expression \( 3x + 5 = 100 \). From here, solving for \( x \) involves simple algebraic manipulation.
The steps include:

• Subtracting 5 from both sides to isolate the term with the variable: \( 3x = 95 \).
• Next, dividing both sides by 3 to solve for \( x \): \( x = \frac{95}{3} \).

Completing these steps, you get \( x \approx 31.67 \), which is the solution to the original equation.
Each step requires careful arithmetic to ensure the accuracy of the result. Double-checking each calculation is a good habit that helps avoid errors. These methods of equation solving are fundamental and widely applicable in various mathematical problems.