Problem 10 - Algebra 2 and Trigonometry

Question

In \(3-14,\) solve each equation for the variable. Express each answer to the nearest hundredth. $$ 5\left(10^{y}\right)=1,200 $$

Step-by-Step Solution

Answer

The solution is approximately \( y = 2.38 \).

Step 1: Isolate the Power

Divide both sides of the equation by 5 to isolate the power of 10.\[ 10^y = \frac{1200}{5} \]This simplifies to:\[ 10^y = 240 \]

Step 2: Take the Logarithm of Both Sides

Apply the logarithm to both sides of the equation to solve for \(y\). Use the base 10 logarithm (common logarithm) because the base of the exponent is 10.\[ \log_{10}(10^y) = \log_{10}(240) \]

Step 3: Apply the Power Rule

According to the power rule for logarithms, \( \log_{10}(10^y) = y \cdot \log_{10}(10) \). Since \( \log_{10}(10) = 1 \), the equation simplifies to:\[ y = \log_{10}(240) \]

Step 4: Calculate the Logarithm

Use a calculator to find \( \log_{10}(240) \). This yields:\[ y \approx 2.3802 \]

Step 5: Round the Solution

Round the result to the nearest hundredth to get the final answer:\[ y \approx 2.38 \]

Key Concepts

Logarithms

Logarithms are mathematical tools that help us to solve equations involving exponential expressions. Essentially, a logarithm is the inverse of taking a power. This means that if you know the result of an exponentiation and the base, you can find what the exponent was. For example, if you know that \(10^y = 240\), using logarithms lets you determine \(y\).

• Common Logarithm: This is a logarithm with base 10, often written as \( \log_{10}\). It is widely used because it relates closely to the way we understand powers of ten, like in the decimal system.
• Power Rule: This rule is vital when working with logarithms. It states \(\log_{b}(x^n) = n \cdot \log_{b}(x)\). So, \(\log_{10}(10^y)\) simplifies to \(y \cdot \log_{10}(10)\), which equals \(y\) because \(\log_{10}(10) = 1\).
• Switching Back: If we have \(y = \log_{10}(240)\), it essentially tells us what exponent we need on 10 to produce 240. Thus, \(10^y = 240\).

When solving equations using logarithms, it is essential to choose the right base that matches the exponents you are dealing with. Often, knowing the properties of logarithms simplifies and speeds up solving equations.

Exponential Functions

Exponential functions involve expressions where the variable is in the exponent. These functions are key in modeling growth and decay processes. Generally, an exponential function can be expressed as \(f(x) = a \cdot b^x\), where \(a\) is a constant, \(b\) is the base of the exponent, and \(x\) is the exponent.

• Base of the Exponent: The base is the number being raised to a power. Common bases include 10 in scientific notation and \(e\) in natural exponential functions.
• Exponent handling: Solving exponential equations often involves isolating the base and using logarithms. For instance, in the equation \(10^y = 240\), we isolate \(10^y\) before applying the logarithm.
• Real-Life Applications: Exponential functions are everywhere. From calculating compound interest to modeling population growth, they play a critical role in understanding dynamic systems. Recognizing the form and properties of exponential expressions allows us greater insight into these processes.

Mastering exponential functions requires understanding how their unique properties affect growth. Once understood, using logarithms to solve these equations becomes much more intuitive.

Rounding Numbers

Rounding numbers is a practical mathematical skill used to decrease the complexity of numbers while retaining their significance to a useful degree. Particularly in the context of logarithmic calculations, rounding ensures that our answers are accurate yet manageable.

• Why Round: Often, results from logarithmic or exponential calculations have many decimal places. Rounding them, such as to the nearest hundredth, keeps computations practical without greatly sacrificing accuracy.
• Steps to Round: Identify the place to round to, then:
  • Look at the digit one place right.
  • If this digit is 5 or greater, increase your rounding place digit by one.
  • If it's less than 5, keep the rounding place digit the same.
• Application Example: In our solution, the number \(y \approx 2.3802\) rounds to \(2.38\) because the third decimal place (0) is less than 5.

Rounding is a key step in presenting clear and concise results, especially in academic and scientific settings where simplifying large or complex numbers helps maintain focus on key findings.