Solving Exponential Equations A Step By Step Guide For 5 * 10^(z/4) = 32
Introduction
In this article, we will delve into the process of solving the equation 5 * 10^(z/4) = 32 for the variable z. This equation involves an exponential term, and our goal is to isolate z and express its solution both as a logarithm in base-10 and as an approximate decimal value rounded to the nearest hundredth. Understanding how to solve exponential equations is crucial in various fields, including mathematics, physics, engineering, and finance. Exponential equations model numerous real-world phenomena, such as population growth, radioactive decay, compound interest, and the spread of diseases. By mastering the techniques to solve these equations, you will gain a valuable tool for analyzing and predicting these phenomena.
This article will guide you through each step of the solution process, providing clear explanations and justifications for each mathematical operation. We will start by isolating the exponential term, then use logarithms to bring the exponent down, and finally, solve for z. We will also discuss the properties of logarithms that are essential for solving exponential equations and provide practical tips for avoiding common mistakes. Whether you are a student learning about exponential equations for the first time or a professional looking to refresh your knowledge, this article will provide a comprehensive guide to solving this type of problem.
So, let's embark on this mathematical journey and unlock the secrets of solving exponential equations! We will begin by revisiting the fundamental concepts of exponential functions and logarithms to ensure we have a solid foundation for tackling the problem at hand. Then, we will methodically work through each step, ensuring a clear and concise understanding of the solution.
Step-by-Step Solution
1. Isolate the Exponential Term
The first crucial step in solving the equation 5 * 10^(z/4) = 32 is to isolate the exponential term. The exponential term in this equation is 10^(z/4). To isolate this term, we need to get rid of the coefficient 5 that is multiplying it. We can achieve this by dividing both sides of the equation by 5. This operation maintains the equality of the equation while bringing us closer to isolating the exponential term.
Dividing both sides of the equation by 5, we get:
(5 * 10^(z/4)) / 5 = 32 / 5
This simplifies to:
10^(z/4) = 6.4
Now we have successfully isolated the exponential term. This step is crucial because it allows us to apply the properties of logarithms in the next step. By isolating the exponential term, we have set the stage for using logarithms to solve for z.
Isolating the exponential term is a fundamental technique in solving exponential equations. It is a necessary prerequisite for applying logarithms, which are the key to unlocking the variable in the exponent. Without isolating the exponential term, we cannot directly apply logarithmic properties to simplify the equation and solve for the unknown variable. This first step is the foundation upon which the rest of the solution is built.
2. Apply Logarithms
With the exponential term isolated, the next step is to apply logarithms to both sides of the equation. Logarithms are the inverse operation of exponentiation, making them the perfect tool for solving exponential equations. The key property we'll use here is that log_b(b^x) = x, where b is the base of the logarithm. In our case, the base of the exponential term is 10, so we will use the base-10 logarithm, commonly denoted as log. Taking the base-10 logarithm of both sides allows us to bring the exponent down and solve for z.
Applying the base-10 logarithm to both sides of the equation 10^(z/4) = 6.4, we get:
log(10^(z/4)) = log(6.4)
Using the logarithmic property log_b(b^x) = x, the left side simplifies to:
(z/4) * log(10) = log(6.4)
Since log(10) = 1, the equation further simplifies to:
z/4 = log(6.4)
This step is crucial because it transforms the exponential equation into a linear equation, which is much easier to solve. By applying logarithms, we have effectively unwrapped the exponent and brought z into a position where it can be readily isolated.
3. Solve for z
Now that we have the equation z/4 = log(6.4), solving for z is a straightforward process. To isolate z, we simply need to multiply both sides of the equation by 4. This will undo the division by 4 on the left side and leave us with z on its own.
Multiplying both sides of the equation by 4, we get:
(z/4) * 4 = 4 * log(6.4)
This simplifies to:
z = 4 * log(6.4)
This is the exact solution for z expressed as a logarithm in base-10. It represents the value of z that satisfies the original equation. However, to get a more intuitive understanding of the value of z, we can approximate it as a decimal.
This step marks the culmination of our algebraic manipulations. We have successfully isolated z and expressed it in terms of a logarithm. This logarithmic form is the precise solution, but for practical applications, we often need a decimal approximation.
4. Approximate the Value of z
To approximate the value of z = 4 * log(6.4), we need to use a calculator to find the value of log(6.4). Most calculators have a base-10 logarithm function, usually denoted as log or log10. Inputting 6.4 into this function will give us the base-10 logarithm of 6.4.
Using a calculator, we find that:
log(6.4) ≈ 0.80618
Now, we multiply this value by 4 to find the approximate value of z:
z ≈ 4 * 0.80618 ≈ 3.22472
Rounding this to the nearest hundredth, we get:
z ≈ 3.22
Therefore, the approximate value of z is 3.22. This means that when z is approximately 3.22, the equation 5 * 10^(z/4) = 32 holds true.
This final step provides us with a practical, numerical answer for z. The approximation allows us to understand the magnitude of z and use it in real-world applications. Rounding to the nearest hundredth provides a balance between accuracy and simplicity.
Solution Summary
In summary, to solve the equation 5 * 10^(z/4) = 32 for z, we followed these steps:
- Isolate the exponential term: Divide both sides by 5 to get 10^(z/4) = 6.4.
- Apply logarithms: Take the base-10 logarithm of both sides to get log(10^(z/4)) = log(6.4), which simplifies to z/4 = log(6.4).
- Solve for z: Multiply both sides by 4 to get z = 4 * log(6.4).
- Approximate the value of z: Use a calculator to find log(6.4) ≈ 0.80618, then multiply by 4 to get z ≈ 3.22472, which rounds to z ≈ 3.22.
Therefore, the exact solution for z is 4 * log(6.4), and the approximate solution rounded to the nearest hundredth is 3.22.
Conclusion
We have successfully solved the exponential equation 5 * 10^(z/4) = 32 for z, expressing the solution both as a logarithm in base-10 (z = 4 * log(6.4)) and as an approximate decimal value rounded to the nearest hundredth (z ≈ 3.22). This process involved isolating the exponential term, applying logarithms to both sides, solving for z, and then approximating the value using a calculator.
Understanding how to solve exponential equations is a fundamental skill in mathematics and has wide-ranging applications in various fields. By mastering this technique, you can tackle more complex problems involving exponential growth, decay, and other related phenomena. The steps outlined in this article provide a clear and concise roadmap for solving similar equations, and the principles discussed can be applied to a variety of mathematical contexts.
Remember, the key to solving exponential equations lies in understanding the relationship between exponential functions and logarithms. By applying the properties of logarithms strategically, you can transform exponential equations into linear equations, making them much easier to solve. Practice is essential for mastering this skill, so try solving other similar equations to solidify your understanding. With consistent effort, you will become proficient in solving exponential equations and gain a valuable tool for tackling mathematical challenges.
In conclusion, the solution to the equation 5 * 10^(z/4) = 32 highlights the power and versatility of logarithms in solving exponential problems. By following the steps outlined in this article, you can confidently approach and solve a wide range of exponential equations, expanding your mathematical toolkit and enhancing your problem-solving abilities.
