Problem 65 - Intermediate Algebra

Question

Solve each equation. Give the exact solution and, when appropriate, an approximation to four decimal places. $$ 7^{x^{2}}=10 $$

Step-by-Step Solution

Answer

Exact: \( x^2 = \frac{\ln(10)}{\ln(7)} \); Approximation: \( x \approx 1.0876 \).

Step 1: Take the Natural Logarithm on Both Sides

Start by taking the natural logarithm on both sides of the equation to bring the exponent down. This gives us:\[\ln(7^{x^2}) = \ln(10)\]Use the property of logarithms that states \( \ln(a^b) = b\ln(a) \).

Step 2: Apply the Power Rule of Logarithms

Using the power rule \( \ln(a^b) = b\ln(a) \), rewrite the left side of the equation:\[x^2 \ln(7) = \ln(10)\]This simplifies the equation to solve for \( x^2 \).

Step 3: Solve for x^2

Divide both sides by \( \ln(7) \) to isolate \( x^2 \):\[x^2 = \frac{\ln(10)}{\ln(7)}\]This is the exact expression for \( x^2 \).

Step 4: Calculate the Approximate Value of x

To find the approximate value of \( x \), first calculate the value of \( x^2 \) using a calculator:\[x^2 \approx \frac{2.3026}{1.9459} \approx 1.183d\]Then, take the square root to find \( x \):\[x \approx \sqrt{1.183}\]Compute accurately to four decimal places to find:\[x \approx 1.0876\]

Key Concepts

Logarithmic Functions

Logarithmic functions play a pivotal role in converting exponential equations into forms that are easier to solve. They are the inverse operations of exponential functions, meaning if you have an equation of the form \(b^x = a\), a logarithmic function can help you find \(x\). Specifically, logarithms allow you to "pull down" an exponent, which is crucial for solving equations where the unknown variable is in the exponent.

When we talk about logarithms, we often refer to two main types: the common logarithm \(\log_{10}\) and the natural logarithm \(\ln\), which we'll discuss later. Logarithmic scales are widely used in various fields, such as calculating the pH in chemistry or measuring the magnitude of earthquakes with the Richter scale. Understanding logarithmic functions is fundamental to grasping many advanced mathematical concepts.

Natural Logarithm

The natural logarithm, denoted as \(\ln\), is a special logarithmic function with the base \(e\), approximately equal to 2.71828. It's called "natural" because it frequently appears in mathematical analysis and is inherently linked to exponential growth and decay processes.

Natural logarithms simplify many complex mathematical expressions. Their key properties make them particularly useful in calculus and mathematical modeling. For example, the natural logarithm of the number \(e\) is 1, since \(\ln(e) = 1\).

• The natural logarithm is useful when solving equations where the variable is in the exponent, like \(7^{x^2} = 10\). We take the \(\ln\) of both sides to "bring down" the exponent, transforming the problem to something more manageable.
• Because \(\ln\) and exponential functions are inverses, \(\ln(e^x) = x\), which is a fundamental identity in calculus and algebra.

Natural logarithms are indispensable in continuous growth modeling, from population dynamics to compound interest calculations.

Power Rule of Logarithms

The power rule of logarithms is an essential property used to handle expressions with exponents. The rule states that \(\ln(a^b) = b\ln(a)\). This rule allows you to transform the power \(b\) into a multiplier, making the equation easier to solve.

For example, in the equation \(7^{x^2} = 10\), applying \(\ln\) and then the power rule gives: \(\ln(7^{x^2}) = x^2 \ln(7)\).

Here's why this is useful:

• By moving the exponent down, you can isolate the variable that was previously stuck in the exponent.
• It simplifies the equation, allowing you to solve for the variable using basic algebraic techniques, such as division and square roots.

Understanding the power rule allows you to solve a wide array of problems involving exponents, making it a powerful tool in both algebra and calculus.