Problem 68 - College Algebra

Question

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

Step-by-Step Solution

Answer

\(t \approx 54.454\)

Step 1: Isolate the Exponential Term

To solve the equation \(1000(1.03)^{t} = 5000\), first isolate the exponential term \((1.03)^{t}\) by dividing both sides of the equation by 1000. This gives:\[(1.03)^{t} = \frac{5000}{1000} = 5.\]

Step 2: Apply the Logarithm

Rewrite the isolated exponential equation \((1.03)^{t} = 5\) using the common log (base 10). Taking the logarithm of both sides, we have:\[\log_{10}((1.03)^{t}) = \log_{10}(5).\]

Step 3: Use the Logarithm Power Rule

Apply the power rule for logarithms, which states that \(\log_{b}(x^a) = a \cdot \log_{b}(x)\). Thus, we rewrite:\[t \cdot \log_{10}(1.03) = \log_{10}(5).\]

Step 4: Solve for the Variable \(t\)

To solve for \(t\), divide both sides of the equation by \(\log_{10}(1.03)\):\[t = \frac{\log_{10}(5)}{\log_{10}(1.03)}.\]

Step 5: Calculate with a Calculator

Use a calculator to find \(\log_{10}(5)\) and \(\log_{10}(1.03)\), and then divide. Approximating to three decimal places, \(\log_{10}(5) \approx 0.69897\) and \(\log_{10}(1.03) \approx 0.012837\). Thus, \(t \approx \frac{0.69897}{0.012837} \approx 54.454\).

Key Concepts

Logarithms

A logarithm is a powerful mathematical tool that helps us analyze exponential relationships. It is essentially the inverse operation to exponentiation. Logarithms allow us to solve for unknown exponents in equations. For instance, in our original exercise, we dealt with an exponential equation: \( (1.03)^t = 5 \). To find the value of \( t \), a logarithm becomes quite handy.
Consider the general definition of a logarithm. If \( a^b = c \), then \( \log_a(c) = b \). This means you ask the question: "To what power must \( a \) be raised, to yield \( c \)?" By applying this idea, we shifted our equation into a solvable format. In the exercise, we used base 10, making it a common logarithm, which you will explore later.
Logarithms aren't limited to base 10; they can have different bases like 2, \( e \) (the natural log), and more. Yet, in practice, base 10 logarithms are prevalent, especially in scientific and financial contexts.

Power Rule for Logarithms

The power rule for logarithms is a fundamental property that simplifies handling exponential expressions. It states that if you have an expression like \( \log_b(x^a) \), it can be rewritten as \( a \cdot \log_b(x) \).
This property makes solving equations involving exponents much easier. You essentially pull the exponent out in front of the log, turning a complex power into a simple multiplier. This was precisely the situation in our original exercise. After applying the common logarithm to each side, the equation \( \log_{10}((1.03)^t) = \log_{10}(5) \) had \( t \) extracted by this rule, transforming the equation to \( t \cdot \log_{10}(1.03) = \log_{10}(5) \).
Remember that this rule is useful for dealing with any exponential expressions in logarithms. It can simplify complex algebraic manipulation and make exponent comparison or solving straightforward.

Common Logarithm

When using logarithms, one might often encounter the concept of 'common logarithm.' The common logarithm involves base 10, and is usually written as \( \log(x) \), without specifying the base explicitly as 10. This is the standard logarithm you would use in several fields such as engineering, science, and even in everyday calculators that usually compute logarithms with this base.
In our initial exercise, we specifically rewrote the equation \( (1.03)^t = 5 \) to everyday terms by applying the common log to both sides. This helps in transforming the exponent problem into a manageable linear equation. Calculators widely prefer common logs because they align with our familiar decimal system.
Using the common log is intuitive when dealing with magnitude, scale, and growth in real-world scenarios. It bridges gap between theoretical mathematics and practical application by offering an accessible approach to understanding and solving exponential equations.