Problem 358 - Precalculus

Question

Solve each equation by rewriting the exponential expression using the indicated logarithm. Then use a calculator to approximate \(x\) to 3 decimal places. \(1000(1.03)^{t}=5000\) using the common log.

Step-by-Step Solution

Answer

The value of \( t \) is approximately 72.896.

Step 1: Isolate the Exponential Term

We start by isolating the exponential expression on one side of the equation. The given equation is \( 1000(1.03)^{t} = 5000 \). To isolate \((1.03)^{t}\), divide both sides of the equation by 1000. This simplifies to: \((1.03)^{t} = \frac{5000}{1000} = 5\).

Step 2: Apply the Common Logarithm

To solve for \( t \), we apply the common logarithm (base 10) to both sides of the equation. This gives us: \( \log((1.03)^{t}) = \log(5) \).

Step 3: Use the Logarithm Power Rule

Utilize the power rule of logarithms, \( \log(a^b) = b\log(a) \), to advance the solution: \( t\log(1.03) = \log(5) \).

Step 4: Solve for t

To isolate \( t \), divide both sides of the equation by \( \log(1.03) \): \( t = \frac{\log(5)}{\log(1.03)} \).

Step 5: Calculate the Value of t

Using a calculator, compute the value of \( t \) by substituting the logs: \( t \approx \frac{\log(5)}{\log(1.03)} \approx 72.896 \). This gives us \( t \approxeq 72.896 \) when rounded to three decimal places.

Key Concepts

Common Logarithm

The common logarithm, often represented as \( \log \), refers to logarithms with a base of 10. This is the type of logarithm that is typically used in many scientific and engineering calculations.

When you see \( \log(x) \) without any base specified, it implies a base of 10. For example, when we compute \( \log(100) \), we are asking, "What power do we need to raise 10 to in order to get 100?" In this case, 10 squared equals 100, so \( \log(100) = 2 \).

Using common logarithms can simplify the process of solving equations involving exponentials by transforming multiplicative relations into additive ones.

• Helps in tackling exponential growth problems.
• Makes it easier to interpret and solve scientific data.
• Used in many fields, including chemistry (pH calculations) and sound (decibels).

In the given exercise, the common logarithm was used to convert the exponential equation \((1.03)^t = 5\) into a manageable form: \( \log((1.03)^t) = \log(5) \). This critical step allows us to further simplify the expression by exploiting another important property of logarithms known as the power rule.

Power Rule of Logarithms

The power rule of logarithms is a valuable tool when dealing with expressions where an unknown variable is an exponent. This rule states that \( \log(a^b) = b \log(a) \).

It's essentially telling us that the exponent on a number can be "brought down" in front of the logarithm, simplifying the expression.

This property is particularly useful because it turns multiplication in the input into multiplication outside, making it easier to tackle non-linear relationships.

• Helps simplify complex exponential equations.
• Provides an approach to isolate the variable of interest.
• Transforms expressions to a form that can be directly evaluated.

In the exercise, applying the power rule transforms \( \log((1.03)^t) = \log(5) \) into \( t \log(1.03) = \log(5) \). This allows us to solve for \( t \) by applying simple algebraic steps, such as division, which are commonplace in logarithmic calculations.

Logarithmic Calculations

Logarithmic calculations are a fundamental part of solving equations involving exponential terms. To solve these equations properly, one generally follows a series of steps involving the properties of logarithms, like the power rule, as discussed earlier.

When performing logarithmic calculations, it's essential to use a calculator because the numbers involved often do not result in neat integer values. Here’s the structured approach:

• Simplify the expression as much as possible before applying any logarithms.
• Apply suitable logarithmic properties (like the power rule) to transform the equation.
• Use a calculator to evaluate logarithmic expressions and approximate values, if necessary.

In the problem context, after simplifying and transforming the expression using logarithmic properties, we arrive at the need to calculate \( t = \frac{\log(5)}{\log(1.03)} \).

This requires using a calculator to determine the numerical values of both \( \log(5) \) and \( \log(1.03) \), and then dividing to find \( t \). The solution yields an approximate value of \( t \approx 72.896 \), rounded to three decimal places.