Problem 35 - Algebra and Trigonometry

Question

Solve each exponential equation . 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. $$ e^{5 x-3}-2=10,476 $$

Step-by-Step Solution

Answer

Therefore, solving the given equation one finds that \(x = \frac{\ln(10,478) + 3}{5}\). Using a calculator for decimal approximation, this result could be further simplified to a number, correct to two decimal places.

Step 1: Isolate the Exponential Part

First, isolate the \(e^{5x-3}\) term in the equation. It can be done by adding 2 to each side of the equation. This provides the relation \(e^{5x-3}=10,478\)

Step 2: Apply the Natural Logarithm

Next, apply the natural logarithm to both sides of the equation. This is done to eliminate the base 'e' in our equation. By the property of logarithms, \(log_a(a^b)=b\), so applying natural log (which is log to base 'e') to our equation simplifies it, resulting in the equation \(5x-3 = \ln(10,478)\)

Step 3: Solve for 'x'

Now, isolate 'x' by adding 3 to both sides of the equation and then dividing by 5. This eventually gives the following equation for 'x': \(x = \frac{\ln(10,478) + 3}{5}\)

Step 4: Approximate the Solution

Use a calculator to find an approximate decimal solution, correct to two decimal places. The exact solution for \(x\) involves the natural log of a non-integer, it can be approximated.

Key Concepts

Natural Logarithms

Natural logarithms, denoted as \(\ln\), are a type of logarithm with the base \(e\), where \(e\) is an irrational constant approximately equal to 2.71828. In the context of solving exponential equations, natural logarithms are incredibly useful because they can be used to ‘undo’ exponents of \(e\). For any positive number \(a\), the natural logarithm of \(a\) answers the question: to what power must \(e\) be raised to produce \(a\)? This is represented mathematically as \(e^{\ln(a)} = a\).

Understanding this concept is crucial for solving exponential equations where the base is \(e\), and it allows us to rewrite the exponent as a simple algebraic expression. In practice, when you apply the natural logarithm to an expression like \(e^{x}\), the base \(e\) and the logarithm cancel each other out, leaving you with just \(x\).

Isolating Variables in Equations

Isolating variables is a fundamental technique in algebra that involves rearranging an equation so that the variable of interest is by itself on one side of the equation. This often requires performing a series of operations that 'undo' what has been done to the variable. In the case of exponential equations, this often means dealing with (and isolating) an exponent that includes the variable.

To isolate a variable, you may need to perform operations such as addition, subtraction, multiplication, division, or even taking roots or applying logarithms, depending on the form of the equation. The goal is to systematically simplify the equation so that you eventually end up with a formula that says 'variable equals some expression,' making it easy to identify the solution.

Properties of Logarithms

Logarithms have several properties that make them useful for solving equations. Three fundamental properties that are often used include:

The Product Rule

This rule states that the logarithm of a product is the sum of the logarithms of the factors: \(\ln(ab) = \ln(a) + \ln(b)\).

The Quotient Rule

Similarly, the logarithm of a quotient is the difference of the logarithms of the numerator and the denominator: \(\ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b)\).

The Power Rule

The logarithm of a power allows the exponent to come down as a multiplier: \(\ln(a^b) = b\cdot\ln(a)\).

These properties are crucial when simplifying complex logarithmic expressions and are instrumental in solving equations that involve exponential terms with variables in the exponent, just like the given exercise.

Calculating Decimal Approximations

Decimal approximations are used when we deal with numbers that are difficult or impossible to represent exactly, such as irrationals or the results of logarithmic functions. Calculators and computers generally approximate such numbers to a certain number of decimal places.

To obtain a decimal approximation for a solution involving natural logarithms, you will typically enter the exact logarithmic expression into a calculator and round to the desired number of decimal places. This is a key step when a problem requires a numeric answer rather than an exact symbolic one. Remember to consult the specific instructions provided, such as rounding to two decimal places, to ensure your final answer is presented correctly.