Question

The value of \(x\) satisfying \(\log _{3} 4-2 \log _{3} \sqrt{3 x+1}=1-\log _{3}(5 x-2)\) (a) 1 (b) 2 (c) 3 (d) none of these

Short answer

Answer: (a) 1

Step-by-step solution

Write down the equation given

We have the equation: \(\log _{3} 4-2 \log _{3} \sqrt{3 x+1}=1-\log_{3}(5 x-2)\)

Apply properties of logarithms

We can use properties of logarithms to simplify our equation: \(\log _{3} \frac{4}{(3x+1)} = \log_{3} \frac{3^1}{(5x-2)}\) Here, we first used \(log_b x − log_b y = log_b(x/y)\) property.

Eliminate logarithms by equating the expressions inside the logs

Since the base of the logarithms is equal, we can now eliminate the logarithms by setting the expressions inside equal to each other: \(\frac{4}{(3x+1)} = \frac{3}{(5x-2)}\)

Solve for x

To solve for x, we can cross-multiply and simplify: \(4(5x-2) = 3(3x+1)\) \(20x - 8 = 9x + 3\) \(11x = 11\) \(x = 1\)

Check the answer

Our solution for x is 1, which matches option (a). So, the correct answer is (a) 1.

Key concepts

Properties of Logarithms

Logarithms are an integral part of algebra and are directly related to exponential functions. Understanding the properties of logarithms is essential for solving equations that involve them.

Key properties include:

• Product Property: This tells us that \(\log_b(MN) = \log_b(M) + \log_b(N)\), which means the logarithm of a product is the sum of the logarithms.
• Quotient Property: Likewise, \(\log_b(\frac{M}{N}) = \log_b(M) - \log_b(N)\) indicates that the logarithm of a quotient is the difference of the logarithms.
• Power Property: It states that \(\log_b(M^k) = k \cdot \log_b(M)\). This property is useful when dealing with logarithms of exponential terms.
• Change of Base Formula: Used to convert logarithms from one base to another: \(\log_b(M) = \frac{\log_c(M)}{\log_c(b)}\), where \(c\) could be any positive value.
• Logarithm of One: For any base \(b\), \(\log_b(1) = 0\) because any number raised to the zero power is one.
• Logarithm of the Base: Another important property is that \(\log_b(b) = 1\), as any base raised to the first power is itself.

By applying these properties astutely, one can transform and simplify logarithmic expressions which make solving equations that contain logarithms much more manageable.

Solving Logarithmic Equations

Solving logarithmic equations often involves applying the properties of logarithms to manipulate and eventually eliminate the logarithmic part of the equation.

Here’s how to approach these problems:

• Firstly, whenever possible, condense multiple logarithmic terms into a single one using the product, quotient, or power properties.
• If you see logarithms on either side of the equation with the same base, you can set the arguments (the values inside the logarithms) equal to each other because if \(\log_b(M) = \log_b(N)\) then \(M = N\).
• If the equation contains a single logarithmic expression equal to a number, you can rewrite the equation as an exponential one, because the logarithm is the inverse of exponentiation. For example, if \(\log_b(x) = a\), then \(x = b^a\).
• If the logarithm bases are different, consider using the change of base formula to make them the same.
• Always check your answers. Logarithmic equations may yield solutions that are not valid in the original equation due to domain restrictions, such as taking the logarithm of a non-positive number.

By incrementally using these strategies, you can dissolve the complexity of logarithmic equations and locate the solution effectively.

Exponential and Logarithmic Functions

Exponential and logarithmic functions are interconnected as they are inverses of each other. An exponential function is of the form \(y = b^x\), where \(b\) is a positive real number, and an increase in \(x\) results in a rapid increase in \(y\). Logarithmic functions, on the other hand, answer the question 'To what exponent must we raise \(b\) to obtain \(x\)?' and is written as \(x = \log_b(y)\).

Their graphs are mirror images if plotted against a diagonal line \(y = x\), which demonstrates their inverse relationship. While exponential functions model growth or decay processes, logarithmic functions are often used to measure the scale of such changes and find the time it takes for a certain level of growth or decay.

Understanding these functions can be crucial in various disciplines including finance for calculating compound interest, in biology for population dynamics, and in earth sciences for measuring the intensity of earthquakes.Moreover, the natural log function \(\ln(x)\) which has \(e\) as its base, is particularly useful in calculus due to the unique properties of \(e\).
The subtleties in handling exponential and logarithmic functions can be essential tools in solving complex real-world problems that involve growth rates and decay.