Problem 53 - Algebra and Trigonometry

Question

Rewrite the equation in terms of base e. Express the answer in terms of a natural logarithm and then round to three decimal places. $$ y=100(4.6)^{x} $$

Step-by-Step Solution

Answer

The given equation in terms of a natural logarithm (base \(e\)) and round to three decimal places (dependent on \(y\)'s value) will be \(x = ln(y/100)/ln(4.6)\).

Step 1: Convert to Logarithmic Form

The exponential equation can be written in logarithmic form using the relationship \(b^{x} = a\) is equivalent to \(x = log_{b}(a)\). Hence, \(y = 100(4.6)^{x}\) can be written as \(x = log_{4.6}(y/100)\).

Step 2: Apply the Base Change Formula

To express this logarithmic function in terms of a natural logarithm (base \(e\)), we apply the base change formula which is \(log_{b}(a) = ln(a)/ln(b)\). Therefore, \(x = log_{4.6}(y/100)\) becomes \(x = ln(y/100)/ln(4.6)\).

Step 3: Round the Final Answer

The final result should be rounded to 3 decimal places. The exact value will depend on the value of \(y\), it can be calculated once \(y\) is known.

Key Concepts

Base Change Formula

When faced with a logarithmic expression or an equation that isn't in the desired base, the base change formula becomes an essential tool. This formula allows us to convert logarithms from one base to another, making it possible to solve equations that originally seem complex.

In mathematical terms, if we need to change the base of a logarithm from base 'b' to a new base 'c', we use the formula: \[ \log_b(a) = \frac{\log_c(a)}{\log_c(b)} \] where 'a' is the value we are taking the logarithm of. This formula is particularly useful when converting to the natural logarithm (base \(e\)), as seen in tasks involving natural logarithms. The base change formula simplifies the transition to base \(e\), which is commonly utilized due to its special properties in calculus and exponential growth scenarios.

It's crucial to note that the base change formula isn't just a mathematical trick, it embodies the proportional relationship between different logarithmic scales and reflects the consistency of logarithmic growth across different bases.

Logarithmic Functions

Logarithmic functions are the inverses of exponential functions, capturing the idea of growth and decay in a profound way. To understand logarithmic functions, one can visualize them as the 'time' it takes for a quantity to reach a certain level under consistent exponential growth (or decay).

The general form of a logarithmic function is: \[ y = \log_b(x) \] This equation tells us that \(y\) is the power to which the base \(b\) must be raised to produce \(x\). When we encounter the natural logarithm, it signifies a base \(e\) logarithmic function, expressed as: \[ y = \ln(x) \] Naturally occurring processes often exhibit exponential patterns, making logarithmic functions vital in fields like biology, economics, and physics. By using these functions, we can unravel the complexities of such growth patterns, revealing the underlying simplicity of nature's framework.

Furthermore, logarithmic functions allow us to address multiplication and division within exponential growth through addition and subtraction, greatly simplifying the mathematical treatment of such phenomena.

Exponential Equations

Exponential equations are foundational in understanding how quantities grow or decay over time at a rate proportional to their size. These equations take the form: \[ y = ab^{x} \] where \(a\) is the initial amount, \(b\) is the base representing the growth or decay factor, and \(x\) is the exponent reflecting the passage of time or the number of growth periods.

In our exercise scenario, we're given the equation \(y=100(4.6)^{x}\), an exponential equation depicting a situation where the quantity \(y\) starts at 100 and grows by a factor of 4.6 for each unit increase in \(x\). These equations are not just theoretical; they model real-world cases like population growth, radioactive decay, and compound interest. They're challenging for their steeply rising or falling curves, which is why converting them into logarithmic form often helps dissect their behavior.

Understanding how to manipulate these equations is crucial in various scientific and financial applications. By converting exponential equations into their logarithmic counterparts, we gain the power to solve for unknowns and predict future outcomes—highlighting their immense value in practical problem-solving.