Problem 11 - Beginning and Intermediate Algebra

Question

Solve the exponential equation algebraically. Then check using a graphing calculator. $$84^{x}=70$$

Step-by-Step Solution

Answer

To solve the exponential equation \(84^x = 70\) algebraically, first take the natural logarithm of both sides: \(\ln(84^x) = \ln(70)\). Then, use the power rule to simplify the left side: \(x\ln(84) = \ln(70)\). To find the value of \(x\), divide by \(\ln(84)\): \(x = \frac{\ln(70)}{\ln(84)}\), which results in \(x \approx -0.19816\). To check this solution using a graphing calculator, graph the functions \(y = 84^x\) and \(y = 70\). The intersection point's x-coordinate should be approximately equal to -0.19816, confirming the solution.

Step 1: Take the logarithm on both sides of the equation

To solve for x, we will use either the natural logarithm (ln) or the logarithm base 10 (log). It doesn't matter which one you choose; we will use the natural logarithm in this case. Take the natural logarithm of both sides of the equation: \[\ln(84^{x})=\ln(70)\]

Step 2: Use the logarithm power rule on the left side of the equation

The power rule states that \(\ln(a^b)=b\ln(a)\). Applying this to the left side of the equation, we get: \[x\ln(84)=\ln(70)\]

Step 3: Solve for x

Divide both sides of the equation by \(\ln(84)\) to isolate x: \[x=\frac{\ln(70)}{\ln(84)}\] Now, use a calculator to calculate the value of x: \[x \approx -0.19816\]

Step 4: Checking the solution using a graphing calculator

To verify our solution using a graphing calculator, graph the two functions \(y=84^x\) and \(y=70\). Observe the point at which the two functions intersect. The x-coordinate of this intersection point should be approximately equal to the value we found for x, -0.19816. If you plot the functions on a graphing calculator, you will find that the intersection point has an x-coordinate of approximately -0.19816, confirming our algebraic solution.

Key Concepts

Logarithmic Functions

In the realm of algebra, logarithmic functions are a vital tool for solving exponential equations. They help us manage numbers and expressions that grow rapidly by converting multiplicative processes into simpler additive ones. In this exercise, logarithms assist in solving the equation \(84^x = 70\).
Logarithms come in multiple bases, including base 10 (represented as \(\log\)) and the natural logarithm base \(e\) (represented as \(\ln\)). Their main purpose is to help us find the exponent when the base and the result are known. For instance, to solve \(84^x = 70\), taking the natural logarithm of both sides transforms it as follows:

• The left side becomes \(\ln(84^x)\)
• The right side becomes \(\ln(70)\)

This use of logarithms is an essential algebraic method for breaking down complex equations to find unknown exponents.

Graphing Calculator

A graphing calculator is immensely helpful for visualizing math problems, especially complex equations involving exponentials and logarithms. In this exercise, it verifies the algebraically determined solution of \(x\) in the equation \(84^x = 70\).
Using a graphing calculator, you can plot the functions \(y = 84^x\) and \(y = 70\). This visual representation allows you to observe where these functions intersect, providing a graphical solution that backs the algebraic one.
When you plot the given functions, the intersection point where \(y = 84^x\) and \(y = 70\) meet will occur approximately at \(x = -0.19816\). This graphical check confirms that the calculated value of \(x\) aligns perfectly with what is shown graphically, reinforcing understanding and accuracy of the solution.

Algebraic Solution

An algebraic solution involves manipulating an equation using algebraic rules and properties to solve for an unknown variable. In the original exercise, we solve the exponential equation \(84^x = 70\) using algebraic steps complemented with logarithms.
Here's the detailed process:

• Start by taking the natural logarithm on both sides, simplifying the exponential equation into a linear form.
• Apply the logarithm power rule to bring the exponent down in front, transforming \(\ln(84^x)\) into \(x \ln(84)\).
• Isolate \(x\) by dividing both sides by \(\ln(84)\).
• Through these steps, you derive \(x = \frac{\ln(70)}{\ln(84)}\), and calculating this gives \(x \approx -0.19816\).

This method provides a precise and logical path to find the solution algebraically, while enhancing conceptual understanding of solving exponential equations.

Power Rule

The power rule for logarithms is a powerful property that simplifies exponential expressions during algebraic manipulations. In logarithmic terms, it states that for any exponential expression \(a^b\), the logarithm can be rewritten as \(b \ln(a)\).
Applying this rule allows us to move the exponent in front of the logarithm, transforming the equation from \(\ln(84^x) = \ln(70)\) to \(x \ln(84) = \ln(70)\).
This transformation is crucial because:

• It converts a complex exponential equation into a linearly solvable form.
• By bringing the exponent down, it makes isolating \(x\) straightforward.

The power rule is not only used here but is an essential concept in calculus as well, showcasing its importance in various mathematical realms.