Solving Exponential Equations 10 * 2^(3t/5) = 1000
In the realm of mathematics, exponential equations hold a significant place, often presenting intriguing challenges to students and enthusiasts alike. These equations, characterized by the presence of a variable in the exponent, require a distinct set of techniques to unravel their solutions. This comprehensive guide delves into the intricacies of solving exponential equations, providing a step-by-step approach that empowers you to tackle these mathematical puzzles with confidence.
Understanding Exponential Equations
Before embarking on the solution process, it's crucial to grasp the essence of exponential equations. An exponential equation is one in which the variable appears as an exponent. A typical exponential equation takes the form:
a oldsymbol{x} = b
where 'a' is the base, 'x' is the exponent (which contains the variable), and 'b' is the result. The key to solving these equations lies in isolating the variable exponent, which can be achieved through a series of algebraic manipulations and the application of logarithmic properties.
Step 1: Isolate the Exponential Term
The first step in solving any exponential equation is to isolate the exponential term. This means manipulating the equation to get the exponential expression by itself on one side of the equation. To isolate the exponential term effectively, follow these steps:
- Identify the exponential term: Pinpoint the term that contains the variable exponent. This is the expression you need to isolate.
- Perform algebraic operations: Employ algebraic operations such as addition, subtraction, multiplication, or division to eliminate any terms that are not part of the exponential expression. The goal is to get the exponential term alone on one side of the equation.
For instance, consider the equation:
In this case, the exponential term is . To isolate this term, we need to divide both sides of the equation by 10:
This simplifies to:
Now, the exponential term is isolated, setting the stage for the next step in the solution process.
Step 2: Apply Logarithms
With the exponential term isolated, the next crucial step involves applying logarithms. Logarithms are the inverse functions of exponentials, and they provide the key to unlocking the variable exponent. To effectively apply logarithms, consider the following:
- Choose the appropriate logarithm base: The choice of logarithm base depends on the base of the exponential term. If the exponential term has a base of 10, use the common logarithm (log base 10). If the exponential term has a base of 'e' (Euler's number), use the natural logarithm (log base e, denoted as ln). If the exponential term has a different base, you can use any logarithm base, but the common or natural logarithm is often preferred for convenience.
- Apply the logarithm to both sides: Take the logarithm of both sides of the equation. This is a crucial step as it allows you to bring the exponent down using logarithmic properties.
- Utilize the power rule of logarithms: The power rule of logarithms states that logb(xm) = m logb(x). This rule is instrumental in solving exponential equations as it allows you to move the exponent from its position as an exponent to a coefficient, making it easier to isolate the variable.
Continuing with our example, we have:
Since the base of the exponential term is 2, we can use any logarithm base. For simplicity, let's use the common logarithm (log base 10). Applying the common logarithm to both sides, we get:
Now, we apply the power rule of logarithms to bring down the exponent:
This transformation is a critical step in solving the equation, as it moves the variable 't' from the exponent to a position where it can be isolated.
Step 3: Solve for the Variable
With the exponent brought down and the variable in a more accessible position, the next step is to solve for the variable. This involves using algebraic manipulations to isolate the variable on one side of the equation. To effectively solve for the variable, consider the following:
- Isolate the term containing the variable: Use algebraic operations such as addition, subtraction, multiplication, or division to isolate the term that contains the variable. The goal is to get the variable term by itself on one side of the equation.
- Solve for the variable: Once the variable term is isolated, perform any remaining algebraic operations to solve for the variable. This may involve dividing both sides by a coefficient or taking the reciprocal of both sides.
Continuing from our previous step, we have:
To isolate the term containing 't', we need to multiply both sides by the reciprocal of , which is :
This simplifies to:
Now, to solve for 't', we divide both sides by log(2):
This expression gives us the solution for 't'.
Step 4: Calculate the Solution
With the variable isolated and expressed in terms of logarithms, the final step is to calculate the solution. This involves using a calculator or logarithmic tables to evaluate the logarithms and perform the necessary arithmetic operations. To accurately calculate the solution, consider the following:
- Evaluate the logarithms: Use a calculator or logarithmic tables to find the values of the logarithms in the expression. Ensure that you are using the correct logarithm base (common or natural logarithm) as per your previous steps.
- Perform the arithmetic operations: Once you have the values of the logarithms, perform the arithmetic operations (multiplication, division, addition, or subtraction) as indicated in the expression. This will give you the numerical value of the solution.
- Round the answer: If necessary, round the answer to the specified decimal place. This is often required in practical applications where an approximate solution is sufficient.
Continuing with our example, we have:
Using a calculator, we find that log(100) ≈ 2 and log(2) ≈ 0.3010. Substituting these values into the expression, we get:
Rounding the answer to the nearest thousandth, we get:
Therefore, the solution to the exponential equation is approximately 11.073.
Example 1
Let's solve the exponential equation: $10 \cdot 2^{\frac{3 t}{5}}=1000$
Step 1: Isolate the Exponential Term
Our first step in solving this exponential equation is to isolate the exponential term. This involves getting the term with the exponent, in this case, , by itself on one side of the equation. We begin with the original equation:
To isolate the exponential term, we need to eliminate the coefficient 10. We do this by dividing both sides of the equation by 10:
This simplifies to:
Now, the exponential term is isolated on the left side of the equation, which sets us up for the next step in the solution process.
Step 2: Apply Logarithms
With the exponential term isolated, the next critical step is to apply logarithms to both sides of the equation. Logarithms are the inverse functions of exponentials, and they are instrumental in solving for variables that appear in exponents. The choice of logarithm base is somewhat arbitrary, but using either the common logarithm (base 10) or the natural logarithm (base e) is often the most convenient, as most calculators readily compute these.
In this case, we will apply the common logarithm (log base 10) to both sides of the equation. This gives us:
Now, we can use the power rule of logarithms, which states that . This rule allows us to bring the exponent down as a coefficient:
Applying the power rule is a key step, as it transforms the exponential equation into a linear equation that we can solve for t.
Step 3: Solve for the Variable
Having applied logarithms and used the power rule, our next goal is to solve for the variable t. We now have the equation:
To isolate t, we need to undo the operations that are being applied to it. First, we multiply both sides of the equation by 5 to eliminate the fraction:
This simplifies to:
Next, we divide both sides by to isolate t:
This simplifies to:
Step 4: Calculate the Solution
Now that we have isolated t, the final step is to calculate the numerical value of the solution. We can use a calculator to find the values of the logarithms and then perform the arithmetic.
Recall that is the logarithm base 10 of 100, which is 2, since . So, we have:
Now, we need to find the value of . Using a calculator, we find that .
Substituting this value, we get:
Rounding this to the nearest thousandth, we get:
Therefore, the solution of the equation , rounded to the nearest thousandth, is approximately 11.073.
Conclusion
Solving exponential equations can seem daunting at first, but by following these four steps – isolating the exponential term, applying logarithms, solving for the variable, and calculating the solution – you can effectively tackle these mathematical challenges. Remember to choose the appropriate logarithm base, utilize the power rule of logarithms, and carefully perform the algebraic manipulations. With practice and a solid understanding of these steps, you'll be well-equipped to conquer exponential equations of varying complexities.