Problem 21 - Precalculus Mathematics for Calculus

Question

Find the solution of the exponential equation, rounded to four decimal places. $$5^{x}=4^{x+1}$$

Step-by-Step Solution

Answer

The solution is approximately 6.2224.

Step 1: Introduce Logarithms

Start by applying the logarithm to both sides of the equation to use the property that allows exponents to be brought down as coefficients. The equation becomes: \[\log(5^{x}) = \log(4^{x+1})\]

Step 2: Apply Logarithmic Power Rule

Use the power rule of logarithms, which states \(\log(a^{b}) = b \cdot \log(a)\), to rewrite both sides:\[x \cdot \log(5) = (x+1) \cdot \log(4)\]

Step 3: Expand and Rearrange Equation

Expand the right side of the equation:\[x \cdot \log(5) = x \cdot \log(4) + \log(4)\]Then, rearrange the terms to get all terms involving \(x\) on one side:\[x \cdot \log(5) - x \cdot \log(4) = \log(4)\]

Step 4: Factor Out x

Factor \(x\) out from the left side of the equation:\[x(\log(5) - \log(4)) = \log(4)\]

Step 5: Solve for x

Divide both sides by \((\log(5) - \log(4))\) to isolate \(x\):\[x = \frac{\log(4)}{\log(5) - \log(4)}\]

Step 6: Calculate x

Use a calculator to compute the logarithms and solve for \(x\). First, calculate \(\log(4)\), \(\log(5)\). Then, substitute these values into the equation:\[x = \frac{\log(4)}{\log(5) - \log(4)} \approx \frac{0.6021}{0.6989 - 0.6021} \approx \frac{0.6021}{0.0968} \approx 6.2224\]

Step 7: Verify and Round the Solution

Verify by substituting back into the original equation to check that both sides are approximately equal. If they are, then round to four decimal places. The solution is already rounded to four decimal places as: 6.2224.

Key Concepts

Logarithms

Logarithms are a mathematical tool used to solve equations involving exponents, especially when variables are in the exponent position. They allow us to transform multiplicative problems into additive ones, simplifying the process. In the equation \(5^x = 4^{x+1}\), applying logarithms to both sides helps us use properties of logarithms to make solving for \(x\) easier.

Key properties of logarithms include:

• The logarithm of a product: \(\log(a \cdot b) = \log(a) + \log(b)\)
• The logarithm of a quotient: \(\log\left(\frac{a}{b}\right) = \log(a) - \log(b)\)
• The logarithm of a power: \(\log(a^b) = b \cdot \log(a)\)

Applying these properties can break down complex exponential expressions into manageable linear equations. By introducing logarithms to both sides of an exponential equation, we create a path to isolate the variable and solve the equation.

Logarithmic Power Rule

The logarithmic power rule is a crucial property that simplifies working with exponents in logarithmic expressions. It states that for any positive number \(a\) and real number \(b\), \(\log(a^b) = b \cdot \log(a)\). This rule allows us to "bring down" the exponent in a logarithmic expression, turning it into a multiplication problem instead.

In our context, for the equation \(\log(5^x) = \log(4^{x+1})\), by applying the power rule, we transform the equation into \(x \cdot \log(5) = (x+1) \cdot \log(4)\).

• Both expressions are simplified by bringing the exponents in front of the logarithms.
• This results in an equation where \(x\) is part of multiplying terms rather than an exponent, simplifying isolation of \(x\).

The logarithmic power rule is essential for tackling exponential equations where isolating the variable in the exponent is necessary for solving the equation.

Solving Equations

Solving equations involves a series of steps aimed at isolating the variable of interest. For exponential equations, this often requires transforming them into linear equations using logarithms. Here's an approach demonstrated in solving \(5^x = 4^{x+1}\):

• Apply logarithms to both sides to make use of logarithmic properties.
• Employ the logarithmic power rule to manage the exponents.
• Rearrange terms to collect like terms on one side of the equation. For example, \(x \cdot \log(5) = x \cdot \log(4) + \log(4)\) becomes \(x \cdot \log(5) - x \cdot \log(4) = \log(4)\).
• Factor out the variable \(x\) from one side. This makes the left hand side of the equation a product of \(x\) and a constant coefficient.
• Divide both sides by this coefficient to solve for \(x\). In this problem, \(x = \frac{\log(4)}{\log(5) - \log(4)}\).

This structured process leads to finding an exact or approximate solution. Calculators are typically required for calculating logarithms to find the numerical solution. Verifying a solution includes substituting it back into the original equation, ensuring both sides are nearly equal, confirming the result. When required, as in this exercise, the solution can be rounded to a specified precision, such as four decimal places.