Question

If \(\log _{a} N=\left(\log _{b} N\right) \times P\), then find \(P\) in terms of \(a\) and \(b:\) (a) \(b^{a}\) (b) \(a^{b}\) (c) \(\log _{a} b\) (d) none of these

Short answer

Answer: (c) \(\log_{a} b\)

Step-by-step solution

Rewrite the equation using change of base formula

We'll rewrite the equation using the change of base formula: \(\log_{a} N = (\log_{b} N) \times P \Rightarrow \frac{\log_{b} N}{\log_{b}a} = (\log_{b} N) \times P\)

Isolate P

Now, let's isolate \(P\) by dividing both sides by \(\log_{b} N\): \(P = \frac{\frac{\log_{b} N}{\log_{b} a}}{\log_{b} N} = \frac{1}{\log_{b} a}\)

Rewrite the expression using the change of base formula

Finally, we will rewrite the expression using the change of base formula again: \(P = \frac{1}{\frac{\log_{a} a}{\log_{a} b}}\)

Simplify the expression

Now we can simplify the expression: \(P = \frac{1}{\frac{1}{\log_{a} b}} = \log_{a} b\) Therefore, the value of \(P\) in terms of \(a\) and \(b\) is \(\log_{a} b\). The correct answer is option: (c) \(\log_{a} b\).

Key concepts

Change of Base Formula

Understanding the change of base formula is essential when working with logarithms. It allows you to convert logarithms from one base to another, which is especially useful if you need to simplify the logarithm or utilize a calculator that doesn't support log calculations in any base other than 10 or e (natural logarithm). The formula is given by:

\[\begin{equation}\log_{a} N = \frac{\log_{b} N}{\log_{b} a}\end{equation}\]
This equates the logarithm of a number N with base 'a' to the quotient of two logarithms: the logarithm of N with base 'b' divided by the logarithm of 'a' with base 'b'. It's like converting the problem into a more familiar or easier-to-solve form. This formula plays a crucial role in reaching the solution for various problems involving logarithms, including our example exercise where it's used to simplify the relationship between \[\begin{equation}\log _{a} N\text{ and } (\log _{b} N) \times P\end{equation}\].
Using this method, you can transform logarithmic equations to solve them using algebra, as illustrated in the solution steps provided.

Isolating Variables

When solving equations, isolating the variable you are solving for is an integral step. It means to manipulate the equation in such a way that you get the target variable on one side of the equation and all other terms on the other side. In the context of logarithmic equations, like the one in our exercise, the process may involve utilizing properties of logarithms or other algebraic techniques such as multiplying, dividing, adding or subtracting across the equation.

To isolate a variable effectively, each step should simplify the equation, making it easier to understand and drawing you one step closer to finding the solution. In the example provided, we isolate the variable \[\begin{equation}P\text{ by dividing both sides of the original equation by } \log_{b} N\text{ which effectively simplifies our problem to a point where } P\text{ stands alone.}\end{equation}\]
The goal of this isolation process is clarity—making it possible to see the solution 'emerge' from the manipulated equation.

Logarithmic Equations

Logarithmic equations involve variables inside a logarithm and require understanding of logarithm properties for their solution. Solving these equations often involves a series of logical and algebraic steps that take advantage of the unique characteristics of logarithms, such as their inverseness with exponentials and their specific rules for multiplication, division, and power. The given exercise is an excellent example of a logarithmic equation where the variable is intertwined within a logarithmic function.

Solving a logarithmic equation typically includes steps like using the change of base formula, combining or expanding logarithms based on log rules, and isolating the variable as seen in the step-by-step solution. Completing these steps allows for the extraction of the variable and ultimately, the solution to the equation.

With such equations, it is crucial to remember that they can only be solved when the arguments of the logarithms are positive, as logarithms of non-positive numbers are undefined. The final solution, \[\begin{equation}P = \log_{a} b\text{ being the correct answer in our problem,}\end{equation}\]shows the proper application of solving strategies and the correct handling of a logarithmic equation.