Problem 40 - Algebra A Combined Function

Question

Solve each equation. Give an exact solution and a four-decimal-place approximation. $$ \log 3 x=1.3 $$

Step-by-Step Solution

Answer

Exact solution: \( x = \frac{10^{1.3}}{3} \); Approximation: \( x \approx 6.6509 \).

Step 1: Understand the Equation Involvement

The given equation is \( \log 3x = 1.3 \). Here, we need to solve for \( x \) such that the logarithmic expression equals 1.3. We will use base 10 logarithms since no base is specified.

Step 2: Isolate the Logarithmic Term

The equation \( \log 3x = 1.3 \) already has the logarithmic term isolated, which means we are set to use properties of logarithms and exponents.

Step 3: Convert the Logarithmic Equation to Exponential Form

Recall that if \( \log_{10} a = b \), then \( a = 10^b \). Here, \( a = 3x \) and \( b = 1.3 \). Therefore, we write the equation in exponential form: \( 3x = 10^{1.3} \).

Step 4: Solve for x

Now we solve for \( x \) by dividing both sides of the equation \( 3x = 10^{1.3} \) by 3, giving \( x = \frac{10^{1.3}}{3} \). Calculate \( 10^{1.3} \) using a calculator, which gives approximately 19.9526.

Step 5: Calculate the Value of x

Thus, \( x = \frac{19.9526}{3} \). Performing the division gives approximately \( x \approx 6.6509 \).

Step 6: Present the Exact and Approximate Solutions

The exact solution in terms of logarithms is \( x = \frac{10^{1.3}}{3} \). The approximate solution to four decimal places is \( x \approx 6.6509 \).

Key Concepts

Exponential Form

Converting a logarithmic equation to exponential form is a vital skill for solving logarithmic expressions. It offers a straightforward way to visualize the problem and simplify the process of finding unknowns. In the given problem, we start with the equation \( \log 3x = 1.3 \). Here is how we can convert it:

• The statement \( \log_{10} a = b \) means \( 10^b = a \).
• This relationship helps us rewrite logarithmic equations as exponential ones, making it easier to solve them.

Applying this to our equation, our logarithmic expression \( \log 3x = 1.3 \) converts to exponential form as \( 3x = 10^{1.3} \). Now, we can easily find the value of \( x \) by calculating the right-hand side and isolating \( x \). This step transforms a potentially complex logarithmic problem into a simple matter of computation and algebraic manipulation.

Base 10 Logarithms

Logarithms are crucial in mathematics because they help simplify multiplications and express large numbers concisely. Specifically, base 10 logarithms, also called common logarithms, are widely used where the base isn't specified explicitly.

• A base 10 logarithm, \( \log_{10} \), measures the power to which the number ten must be raised to obtain a given number.
• In our example, the expression \( \log 3x = 1.3 \) is understood to be in base 10.

The notion behind the equation is that 10 raised to the power of 1.3 will equal the number 3x. Recognizing when to use base 10 logarithms is important as it often simplifies calculations, especially when calculators are involved in finding approximations or dealing with scientific notation. Using calculators that handle base 10 logarithms can facilitate the computation of the equation in a step-by-step manner.

Exact and Approximate Solutions

When solving logarithmic equations, providing both exact and approximate solutions highlights the precision of computations and the practicality of the results.

• Exact solutions are the most precise form, expressed algebraically. For example, in this exercise, the solution \( x = \frac{10^{1.3}}{3} \) represents the exact value.
• Approximate solutions are numerical estimates that make it practical to understand and use in real-world applications. In this case, we approximate and say \( x \approx 6.6509 \).

Exact solutions are crucial for theoretical exploration and providing insight into the nature of the equation. However, approximate solutions are often more user-friendly and tangible, particularly when addressing day-to-day scenarios or further predictions. Calculators are often employed for these approximations and can easily express these numbers in decimals for clear communication and application in various contexts.