Problem 30 - Elementary and Intermediate Algebra

Question

Solve each equation. Give the exact solution and an approximation to four decimal places. See Example 3. $$ 7^{x}=12 $$

Step-by-Step Solution

Answer

The exact solution is \(x = \frac{\ln(12)}{\ln(7)}\), approximately 1.2765.

Step 1: Take the Logarithm of Both Sides

To solve the equation \(7^x = 12\), we first take the logarithm of both sides. This allows us to bring the exponent down, making it easier to solve for \(x\). We can use either the natural logarithm \(\ln\) or the common logarithm \(\log\). For this problem, we'll use the natural logarithm, \(\ln\):\[ \ln(7^x) = \ln(12) \]

Step 2: Apply the Power Rule of Logarithms

Using the property of logarithms, \(\ln(a^b) = b \cdot \ln(a)\), we can simplify \(\ln(7^x)\) to \(x \cdot \ln(7)\):\[ x \cdot \ln(7) = \ln(12) \]

Step 3: Solve for x

To isolate \(x\), divide both sides by \(\ln(7)\):\[ x = \frac{\ln(12)}{\ln(7)} \]

Step 4: Calculate the Exact Value of x

Using a calculator, find the values of \(\ln(12)\) and \(\ln(7)\), then divide them to find \(x\):\[ x = \frac{\ln(12)}{\ln(7)} \approx 1.2765 \]

Key Concepts

Exponent Rules

Exponent rules are essential when dealing with expressions that involve powers. They provide straightforward methods of manipulating equations to extract meaningful solutions. Here’s what you need to know about exponent rules:

• The Product Rule of Exponents states that when you multiply two exponential terms with the same base, you add the exponents: \(a^m \cdot a^n = a^{m+n}\).
• The Quotient Rule indicates that when you divide two exponential terms with the same base, you subtract the exponents: \(\frac{a^m}{a^n} = a^{m-n}\).
• The Power Rule for Exponents specifies that when raising an exponent to another power, you multiply the exponents: \((a^m)^n = a^{m \cdot n}\).

Using these rules allows simplification of equations for easier manipulation. In the problem \(7^x = 12\), understanding exponent rules made it easier to transform the equation using logarithms.

Natural Logarithm

The natural logarithm is a powerful tool in algebra, represented as \(\ln\). It is the logarithm to the base \(e\), an irrational constant approximately equal to 2.71828. Natural logarithms are particularly useful in calculus and when solving exponential growth problems.

• The natural logarithm of a number answers the question: To what power must \(e\) be raised to produce that number?
• In solving equations like \(7^x = 12\), the natural logarithm helps by bringing the exponent \(x\) down, using the logarithmic property \(\ln(a^b) = b \cdot \ln(a)\). This transformation converts the problem into a linear equation, which is easier to solve.

By employing the natural logarithm, as shown in our solution, we simplified the problem to \(x \cdot \ln(7) = \ln(12)\), making the isolation of \(x\) a simple division.

Power Rule of Logarithms

The power rule of logarithms is an essential property that simplifies the evaluation of logarithms involving exponents. This rule states:
\[\ln(a^b) = b \cdot \ln(a)\]

• This rule allows you to move the exponent in front of the logarithm, effectively converting an exponential expression into a linear one.
• Using the power rule means that instead of handling complex exponentials directly, we work with simpler and more familiar arithmetic.
• It is particularly useful when attempting to isolate a variable in an exponent, simplifying both logarithmic and exponential equations efficiently.

In our problem, applying the power rule allowed us to turn \(\ln(7^x)\) into \(x \cdot \ln(7)\), directly facilitating the solution process by converting it into a solvable arithmetic equation: \(x = \frac{\ln(12)}{\ln(7)}\).