Problem 20 - Algebra and Trigonometry

Question

Find the solution of the exponential equation, rounded to four decimal places. \(\left(\frac{1}{4}\right)^{X}=75\)

Step-by-Step Solution

Answer

The solution is approximately \( X = -3.1145 \).

Step 1: Express the Equation in Logarithmic Form

First, apply logarithms to both sides of the equation to solve for \( X \). The given equation is \( \left( \frac{1}{4} \right)^X = 75 \). Taking the logarithm of both sides gives us \( \log_{\left( \frac{1}{4} \right)} 75 = X \).

Step 2: Convert to Common Logarithms

Convert the logarithmic equation into a form using common logarithms or natural logarithms. Using the change of base formula, we use the base 10 logarithm: \( X = \frac{\log_{10} 75}{\log_{10} \left( \frac{1}{4} \right)} \).

Step 3: Calculate the Logarithms

Use a calculator to find the values of the logarithms. Calculate \( \log_{10} 75 \approx 1.8751 \) and \( \log_{10} \left( \frac{1}{4} \right) \approx -0.6021 \).

Step 4: Solve for X

Substitute the logarithmic values into the equation from Step 2: \( X = \frac{1.8751}{-0.6021} \). Calculate \( X \approx -3.1145 \).

Step 5: Round the Solution

Round the number to four decimal places, which gives us \( X \approx -3.1145 \) as the final answer.

Key Concepts

Logarithmic Form

The logarithmic form is a way to express exponential equations as logarithms. This form helps in solving exponential equations by making it easier to manipulate the terms. When you have an exponential equation like \[\left( \frac{1}{4} \right)^{X}=75\] you can convert it into logarithmic form to help isolate the variable \(X\). The general idea is that if \[b^X = a\] then you can write \[\log_b a = X.\] Thus, the exponential equation \[\left( \frac{1}{4} \right)^X = 75\] becomes \[\log_{\left( \frac{1}{4} \right)} 75 = X.\] This conversion is an essential first step in solving exponential equations using logarithms. It allows the equation to be rewritten in a form that can be closely examined using various types of logarithms.

Change of Base Formula

The change of base formula is a helpful tool for working with logarithms, especially when specific logarithmic bases are not easily accessible with simple calculators. The formula is as follows:\[\log_b a = \frac{\log_c a}{\log_c b}.\]This allows you to convert a logarithm in one base to another base that is more convenient, such as base 10 (common logarithms) or base \(e\) (natural logarithms). For our problem, aiming to solve the equation by using the change of base formula, we can transform:\[X = \log_{\left(\frac{1}{4}\right)} 75\]to:\[X = \frac{\log_{10} 75}{\log_{10} \left( \frac{1}{4} \right)}.\]This conversion simplifies calculation because base 10 logarithms are usually available on most calculators. It's a key step in practical applications of logarithmic equations in real-life scenarios.

Common Logarithms

Common logarithms are logarithms with base 10. They are denoted as \( \log_{10} \), though sometimes the base is omitted because it is so commonly used. When solving equations, common logarithms provide a convenient way to handle calculations, as many calculators are equipped with a dedicated \( \log \) button for computing base 10 logarithms.In our equation, converting the logarithmic form using base 10 logarithms makes the calculation straightforward. We compute \( \log_{10} 75 \) to approximately 1.8751, which is vital for solving the equation using common logarithms.- A simple conversion and access through calculators make them a popular choice.- Facilitates quick and easy arithmetic operations.By understanding common logarithms, you can solve various equations more efficiently, making them an essential tool in mathematical problem-solving.

Natural Logarithms

Natural logarithms use the base \(e\), where \(e\) is an irrational constant approximately equal to 2.71828. They are denoted as \(\ln\). Natural logarithms are particularly prevalent in calculus and complex mathematical computations because of their natural base, \(e\).Although base 10 logarithms were used in solving our particular exponential equation, natural logarithms could equally serve the purpose using:\[\log_{\left(\frac{1}{4}\right)} 75 = \frac{\ln 75}{\ln \left( \frac{1}{4} \right)}.\]This alteration demonstrates the flexibility with which we can approach logarithmic calculations using different bases.- Natural logarithms are crucial in higher mathematics for solving growth models, decay processes, and more.- They provide insights into patterns found within nature due to the special properties of \(e\).Mastering natural logarithms will give you another strong tool in your toolbox for tackling complex mathematical challenges.