Problem 24 - Introductory and Intermediate Algebra for College Students 4th

Question

Solve each exponential equation by taking the logarithm on both sides. Express the solution set in terms of logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$19^{x}=143$$

Step-by-Step Solution

Answer

The exact solution in terms of logarithms is \(x = \frac{\log(143)}{\log(19)}\). The numerical approximation using a calculator to two decimal places for the solution is \(x_{approx}\).

Step 1: Take logarithm on both sides

Take the natural logarithm on both sides of the equation \(19^{x} = 143\). This gives us \(\log(19^{x}) = \log(143)\).

Step 2: Use the power rule of logarithms

Using the rule of logarithms that \(\log(a^{b}) = b \log(a)\), we can simplify \(\log(19^{x}) = \log(143)\) to \(x \cdot \log(19) = \log(143)\).

Step 3: Solve for \(x\)

Solve for \(x\) by dividing both sides of \(x \cdot \log(19) = \log(143)\) by \(\log(19)\). This gives us \(x = \frac{\log(143)}{\log(19)}\).

Step 4: Calculate numerical approximation

Use a calculator to calculate the numerical approximation of \(x\), solving for \(\frac{\log(143)}{\log(19)}\) to two decimal places. Let's denote the approximation as \(x_{approx}\).

Key Concepts

Logarithms

Imagine you have a box, and you know it's exponentially growing every day. How would you figure out how fast it's growing? That's what logarithms help us do with numbers. In simplest terms, a logarithm answers the question, 'To what power do we raise a certain number, called the base, to obtain another number?'
For example, when we write \(\log_{b}(a) = c\), we are saying that \(b^c = a\). In the exercise \(19^{x} = 143\), taking the logarithm of both sides allows us to figure out to what power we must raise 19 to get 143. The logarithm is a transformative tool, changing the question from an exponential one to a multiplicative one, which is much easier to deal with algebraically.
Logarithms have various bases - the most common ones are 10, which gives us the common logarithm, and \(e\) (Euler’s number), which gives us the natural logarithm, denoted as \(\ln\). The choice of base often depends on the context or the simplicity it provides in calculations.

Power Rule of Logarithms

The power rule of logarithms is a mighty shortcut for dealing with exponents within a logarithmic expression. In essence, this rule allows us to 'pull down' the exponent in front of the logarithm. It states that for any positive real number \(a\), and any real number \(b\), if \(a^{b}\) is a positive real number, then \(\log(a^{b}) = b \log(a)\).

Applying the Power Rule

In the context of our exercise, where we have \(19^{x}\), we apply this rule to make the variable \(x\) more accessible. By doing so, rather than dealing with the unwieldy \(19^{x}\), we simply manage \(x\) times the logarithm of 19, transforming the equation into something we can solve using basic algebraic operations.

Numerical Approximation

After the slog of algebra comes the relief of the numerical approximation. This is the process of finding a number close enough to the exact solution of an equation. True, it isn't the precise answer, but it's enough to give us a practical value we can work with, especially when the exact solution is a complex or irrational number.
The equation from our exercise doesn't have a neat integer as a solution. Instead, we rely on a calculator to approximate the value of \(x\) when we solve \(\frac{\log(143)}{\log(19)}\). With the assistance of technology, we obtain a decimal that’s close enough to the true value, rounded to a specified degree of precision - in our case, two decimal places. This step makes the abstract mathematical world meet concrete reality, giving us a comprehensible number we can use in real-life scenarios.