Problem 38 - College Algebra

Question

Solve each exponential equation in Exercises \(23-48\). Express the solution set in terms of natural logarithms or common logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$ 5^{x-3}-137 $$

Step-by-Step Solution

Answer

The solution in terms of logarithms is \(\log_{5}(137) + 3 = x\). Use a calculator to find an approximate decimal equivalent, to two decimal places.

Step 1: Isolate the exponential term

This is done by adding \(137\) to both sides of the equation to isolate the exponential term \(5^{x-3}\) on one side: \[5^{x-3} = 137\]

Step 2: Converting to a logarithm

Convert to logarithmic form. This will make the equation easier to solve. To do this, we recognize that \(a = b^c\) is equivalent to \(log_b(a) = c\). Thus:\[\log_{5}(137) = x-3\]

Step 3: Solve for x

Next, solve for \(x\) by adding \(3\) to both sides of the equation:\[\log_{5}(137) + 3 = x\]. Now, \(x\) has been isolated.

Step 4: Calculate the decimal approximation

Finally, calculation of the decimal approximation could be done using a calculator. It's important to round to two decimal places per the original exercise.

Key Concepts

Natural Logarithms

When we work with equations involving exponents, natural logarithms are an incredibly useful tool for finding solutions. A natural logarithm, denoted as \(\ln\), is a logarithm to the base \(e\), where \(e\) is an irrational and transcendental number approximately equal to 2.71828. It often appears in problems involving continuous growth or decay, like interest calculations and natural processes. To use a natural logarithm to solve an exponential equation like \(a^x = b\), you would take the natural log of both sides. The equation then simplifies to \(x \ln(a) = \ln(b)\), because the natural logarithm is the inverse operation to the exponential function with base \(e\). With this transformation, you can solve for \(x\) more easily.

Common Logarithms

Common logarithms are similar to natural logarithms, but instead of using the base \(e\), they use the base 10. Common logarithms are usually denoted as \(\log\) without a base specified, because base 10 is so commonly used it's assumed. When you encounter an exponential equation such as \(10^x = 100\), you could use common logarithms and the equation becomes \(\log(100) = x\). If you're solving an equation with a base other than 10, you might use the change of base formula, \(\log_b(a) = \frac{\log(a)}{\log(b)}\), to convert it into an expression involving common logarithms. This could then be approximated using a calculator.

Exponential Function

An exponential function is defined by an equation where the variable appears as an exponent, as in \(f(x) = a^x\), where \(a\) is a positive real number. The graph of an exponential function shows rapid growth or decay and is crucial in modeling situations where things increase or decrease at a rate proportional to their current value, such as population growth or radioactive decay. Solving an exponential equation typically involves isolating the exponential term and then applying a logarithm to both sides of the equation to bring the exponent down, effectively linearizing the equation and making \(x\) solvable.

Decimal Approximation

Often, when solving exponential equations, you'll end up with an answer involving logarithms that doesn't simplify to a nice whole number; hence the need for decimal approximation. A decimal approximation is a way to express these complex results as rounded decimal numbers that are easier to understand and use in practical situations. Using a calculator, you can find the decimal approximation of any logarithm. For the problem \(\log_{5}(137) + 3 = x\), you would first calculate \(\log_{5}(137)\) using the change of base formula and then add 3 to get \(x\). Rounding to two decimal places provides a solution that is both accurate and useful.