Question

If \(\log _{16} \mathrm{x}=2.5\), then \(\mathrm{x}=\) (1) 40 (2) 256 (3) 1024 (4) None of these

Short answer

Question: Given \(\log _{16} x=2.5\), find the value of x from the following options: (1) 64 (2) 625 (3) 841 (4) None of these Answer: (4) None of these Explanation: We have converted the logarithm equation into an exponential form and calculated the value of x. By comparing the value of x with the given options, we can determine that the correct answer is (4) None of these since x is approximately equal to 843.75.

Step-by-step solution

Write down the given logarithm equation

We are given the logarithm equation: \(\log _{16} x=2.5\)

Convert the logarithm equation into an exponential form

We can rewrite the logarithm equation in exponential form by using the definition of logarithms: \(b^{\log_{b}a} = a\). So, we have: \(16^{2.5} = x\)

Calculate the value of x

Now, we need to calculate the value of x using the exponential equation: \(x = 16^{2.5}\). We can rewrite \(16^{2.5}\) as \((16^2)^{1.25}\) and as we know that \(16^2 = 256\), thus we will have: \(x = 256^{1.25}\)

Compare the calculated value of x with given options

Now, we have found the value of x: \(x = 256^{1.25}\) By calculating this value we can roughly estimate the value, that lies between the options (2) and (3). With further calculation we can identify that the value is approximately equal to 843.75. Hence, the correct answer is (4) None of these.

Key concepts

Exponential Form

When we talk about exponential form, we are describing a way to express numbers using powers. It's a fundamental concept in mathematics that relates directly to logarithms. Given a logarithmic equation, the exponential form helps us find the magnitude of a number by raising a base to a certain power. For instance, in the equation given in the exercise \( \log_{16} x = 2.5 \), the exponential form is obtained by rewriting it as \( \16^{2.5} = x\).
This conversion uses the key principle that \( b^{\log_{b}a} = a\). This simply means that if you take the base (in this case, 16) and raise it to the power given by the logarithm (2.5), you'll get the number (x). It's crucial to understand this conversion because it breathes life into the more abstract concept of logarithms, turning them into actionable calculations. What’s fascinating is that exponential form lays the groundwork for solving the equation, as it allows mathematical operations to proceed with more straightforward arithmetic treatments.

Logarithmic Equation

A logarithmic equation involves an unknown quantity within the logarithm itself. It's essentially an equation that links the logarithm of a number in a specific base to an exponent. In our original exercise, we deal with \( \log_{16} x = 2.5 \). This tells us that 16 needs to be raised to an exponent of 2.5 to equal some number x.
One of the main purposes of solving a logarithmic equation is to determine this unknown number. The definition of a logarithm is that it answers the question: to what power must we raise a given base to obtain a certain value? By manipulating a logarithmic equation, especially converting it into an exponential form, we uncover the inherent problem, and it leads us to solve for the unknown. Understanding this basic framework allows students to break down more complex logarithmic equations methodically.

Calculation of Logarithms

Calculating logarithms means determining the power a base must be raised to yield a particular number. In practical terms, when we are given the problem \( \log_{16} x = 2.5 \), the calculation process involves converting this into a computation-ready format, i.e., its exponential equivalent.
Once in exponential form \( \16^{2.5} = x\), the calculation flows more naturally. Since \( 16^{2.5} \), equals \( 256^{1.25} \), you begin by calculating each exponentiated component to arrive at approximately 843.75. This kind of calculation, while straightforward in process, crucially depends on understanding the base and exponent correctly. Automating these calculations often involves using a calculator or computational tool, especially when numbers are not whole numbers. It emphasizes understanding how to decompose and apply exponential rules to evaluate logarithmic expressions accurately. Understanding the calculation helps verify answers and choose correct options from those provided in exercises like the one given.